The Interplay of System Dynamics and Dry Friction ... · The Interplay of System Dynamics and Dry Friction: Shakedown, Ratcheting and Micro-Walking FG Systemdynamik und Reibungsphysik

The Interplay of System Dynamics and Dry Friction:

Shakedown, Ratcheting and Micro-Walking

FG Systemdynamik

und Reibungsphysik

vorgelegt von

Dipl.-Ing. Robbin Wettergeb. in Berlin

von der Fakultat V Verkehrs- und Maschinensystemeder Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss

Vorsitzender: Prof. Dr.-Ing. Christian Oliver PaschereitGutachter: Prof. Dr. rer. nat. Valentin L. PopovGutachter: Prof. Dr.-Ing. habil. Manfred ZehnGutachter: Dr. Sc. Alexander Filippov

Tag der wissenschaftlichen Aussprache: 27. Januar 2016

Berlin 2016

Acknowledgements

This thesis was created during my work as a scientific assistant at the Department ofSystem Dynamics and Friction Physics at the Institute of Mechanics at the TechnischeUniversitat Berlin. I am especially grateful to my supervisor and chair of the departmentProf. Valentin Popov for his guidance and continued support. His encouragements andhis many helpful suggestions as well the great freedom during my work were essentialto me. I also thank Prof. Manfred Zehn and Prof. Alexander Filippov for their workas referees in the doctoral committee and many valuable discussions. Their quick as-sessment was a great help in the special situation after the death of Professor ZygmuntRymuza in December 2015. I am also grateful to Prof. Christian Paschereit for takingthe chair of the doctoral committee.

The ten years at the department were a very special time for me and I am very gratefulfor having met so many inspiring colleagues and friends. In particular, I would like tothank Roman Pohrt for his willingness to discuss so many ideas, Tobias Rademacherfor his thorough proofreading and Jasminka Starcevic for her help in the laboratory.Last but not least I would like to thank my wonderful wife Miriam for her patience andsupport without which this work would not have been possible.

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Abstract

This thesis discusses the influence of vibrations in dynamic systems with dry friction.Force locked connections such as screws, bolts and interference fits are in practice oftensubject to external vibrations. This may lead to an incremental slip of the contactinterface and failure of the connection, even if the tangential force is insufficient tocause complete sliding. Vibrations also affect systems with gross sliding such as brakes,clutches and joints. Their frictional resistance strongly depends on system dynamics andits corresponding variables such as mass, velocity and stiffness.As a generic model for a force locked connections under the influence of vibrations, the socalled oscillating rolling contact is introduced. It consists of a tangentially loaded contactbetween two bodies, where the upper body follows an oscillating rocking motion. Forthe analysis the well-known Method of Dimensionality Reduction (MDR) is used andthree-dimensional simulations are performed. In addition, an experimental rig is usedfor comparison. The results show that the oscillatory rolling influences the pressuredistribution and the contact region. Together with the tangential load, this rockingmotion causes incremental sliding processes and a macroscopic rigid body motion. In casethat the rolling amplitude is sufficiently small, the slip ceases after the first few periods.The residual force in the contact withstands the tangential load and the system reaches anew equilibrium: a so called shakedown occurs. Otherwise a continuing macroscopic rigidbody motion is induced where one side of the contact alternately sticks, while the otherslips. This effect is referred to as ratcheting. Using the MDR analytical failure criteria,i.e. the so called shakedown limits, are derived as a function of the rolling amplitudeand the tangential load. These are in very good agreement with the experimental resultsand the three-dimensional simulations. Furthermore, the incremental shift per period incase of ratcheting is determined.In order to examine the influence of system dynamics on sliding friction, the micro-walking machine is introduced as a model. It consists of a rigid body with a numberof elastic contact spots that is pulled by a constantly moving base. In addition to anexperimental rig, numerical integration and an extensive parameter study are used forthe analysis. The results show that the frictional resistance almost vanishes if a certainparameter range is met. If so, self-excited oscillations occur that lead to a particulardynamic mode where the motion is characterized by a strong correlation between low oreven zero normal forces and a fast forward motion. This effect is referred to as micro-walking and leads to a theoretical reduction of the frictional resistance of up to 98 %.These results are confirmed by the experiments where the maximal reduction is 73 %.Following the analysis and identification of the relevant parameter ranges for all of thethree effects, their occurrence in technical systems discussed. In addition guidelines aregiven for possible technical applications.

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Kurzzusammenfassung

Diese Arbeit beschreibt den Einfluss von Vibrationen in dynamischen Systemen mittrockener Reibung. Kraftschlussige Verbindungen wie Schrauben, Bolzen und Press-passungen sind in der Praxis haufig externen Vibrationen ausgesetzt. Dies kann zueinem inkrementellen Schlupf in der Kontaktflache und einem Versagen der Verbindungfuhren, auch wenn die Tangentialkraft nicht ausreicht, vollstandiges Gleiten zu erzeu-gen. Vibrationen beeinflussen ebenso das Verhalten von Systemen, in denen komplettesGleiten auftritt. Beispiele hierfur sind Bremsen, Kupplungen und Gelenke. Deren Reib-widerstand hangt stark von der Systemdynamik und den Einflussgroßen wie Masse,Geschwindigkeit oder Steifigkeit ab.Als allgemeines Modell einer kraftschlussigen Verbindung unter dem Einfluss von Vibra-tionen dient der sogenannte oszillierende Rollkontakt. Dieser besteht aus einem tangen-tial belasteten Kontakt zweier Korper, wobei der obere eine oszillierende Wipp-Bewegungausfuhrt. Fur die Analyse werden die Methode der Dimensionsreduktion (MDR) ver-wendet und dreidimensionale Simulationen durchgefuhrt. Ein Versuchsstand dient zurVerifikation. Es zeigt sich, dass das oszillierende Rollen die Druckverteilung und dieKontaktflache variiert. Zusammen mit der tangentialen Belastung erzeugt dies ein inkre-mentelles Gleiten und eine makroskopische Starrkorperbewegung. Im Falle einer ausrei-chend kleinen Rollamplitude stoppt das Gleiten nach einigen Perioden. Die Restkraftim Kontakt widersteht der tangentialen Belastung und das System erreicht einen neuenGleichgewichtszustand: ein sogenannter Shakedown tritt auf. Andernfalls wird eine fort-gesetzte Starrkorperbewegung induziert, bei der abwechselnd eine Seite des Kontakteshaftet, wahrend die andere gleitet. Dies wird als Ratcheting bezeichnet. Mit Hilfe derMDR werden Versagenskriterien als Funktion der Rollamplitude und der Tangentialkrafthergeleitet. Diese sogenannten Shakedown-Grenzen stimmen sehr gut mit den experi-mentellen Ergebnissen und den dreidimensionalen Simulationen uberein. Zusatzlich wirdfur den Fall des Ratcheting die inkrementelle Verschiebung bestimmt.Zur Untersuchung des Einflusses der Systemdynamik auf die Gleitreibung dient diesogenannte Mikro-Laufmaschine. Diese besteht aus einem starren Korper mit einerAnzahl von elastischen Kontaktstellen, der von einem konstant bewegten Fußpunktgezogen wird. Neben einem Versuchsstand werden zur Analyse numerische Integrationund umfangreichen Parameterstudien verwendet. Die Ergebnisse zeigen, dass der Rei-bungswiderstand fast verschwindet, wenn ein bestimmter Parameterbereich eingestelltwird. Es treten selbsterregte Schwingungen auf, die zu einem charakteristischen dy-namischen Modus fuhren. Die Bewegung ist durch eine starke Korrelation zwischenniedrigen oder verschwindenden Normalkraften und einer schnellen Vorwarts-Bewegunggekennzeichnet. Dieses sogenannte Micro-Walking fuhrt zu einer theoretischen maxi-malen Reduzierung des Reibungswiderstandes von bis zu 98 %. Dies konnte experi-mentell bestatigt werden, wobei die maximale Reduktion im Experiment bei 73 % lag.Nach der Analyse und der Identifizierung der relevanten Parameterbereiche fur alle dreiEffekte wird ihr Vorkommen in technischen Systemen diskutiert. Daruber hinaus werdenLeitlinien fur mogliche technische Anwendungen gegeben.

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iv

Contents

1 Introduction 1

1.1 State of Scientific Research . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Nominally Static Systems . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Fundamentals and Methods 9

2.1 Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Spherical Normal Contact . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Spherical Tangential Contact . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Steady State Rolling Contact . . . . . . . . . . . . . . . . . . . . . 14

2.2 The Method of Dimensionality Reduction . . . . . . . . . . . . . . . . . . 16

2.2.1 Spherical Tangential Contact using MDR . . . . . . . . . . . . . . 18

2.2.2 Steady-State Rolling Contact using MDR . . . . . . . . . . . . . . 20

2.3 Modelling of Frictional Systems . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Separation of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Modelling of Coulomb Friction in Dynamic Systems . . . . . . . . 24

2.3.3 Algorithm for the Modelling of Contacts . . . . . . . . . . . . . . . 25

3 Shakedown and Ratcheting in Incomplete Contacts 27

3.1 Oscillating Rolling Contact . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Influence of the Oscillating Rolling . . . . . . . . . . . . . . . . . . 29

3.1.2 MDR Model of the Oscillating Rolling Contact . . . . . . . . . . . 30

3.2 Shakedown of the Oscillating Rolling Contact . . . . . . . . . . . . . . . . 31

3.2.1 Contact Configuration after Shakedown . . . . . . . . . . . . . . . 34

3.2.2 Shakedown Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Ratcheting of the Oscillating Rolling Contact . . . . . . . . . . . . . . . . 40

3.4 Influence of the Alignment of Load and Oscillation . . . . . . . . . . . . . 42


3.4.2 Shakedown Limits for the Perpendicular Case . . . . . . . . . . . . 44

3.4.3 Ratcheting for the Perpendicular Case . . . . . . . . . . . . . . . . 47

3.5 Oscillating Cylindrical Rolling Contact . . . . . . . . . . . . . . . . . . . . 49

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3.5.1 Influence of the Oscillating Rolling . . . . . . . . . . . . . . . . . . 51


3.5.3 Shakedown of the Cylindrical Rolling Contact . . . . . . . . . . . . 56

3.5.4 Ratcheting of the Cylindrical Rolling Contact . . . . . . . . . . . . 58

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Dynamic Influences on Sliding Friction 61

4.1 The Micro-Walking Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 Discrete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Parameters of Influence . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.3 Modelling and Simulation . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 System Dynamics and Frictional Resistance . . . . . . . . . . . . . . . . . 69

4.2.1 Influence of the Parameters . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2 Reduction of the Frictional Resistance . . . . . . . . . . . . . . . . 74

4.2.3 Vanishing Frictional Resistance . . . . . . . . . . . . . . . . . . . . 78

4.2.4 Summary of the Theoretical Results . . . . . . . . . . . . . . . . . 81

4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.2 Comparison of Experiment and Model . . . . . . . . . . . . . . . . 85

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Experimental Analysis 89

5.1 Experimental Rig for the Rolling Body Effects . . . . . . . . . . . . . . . 89

5.1.1 Parallel Alignment of Force and Oscillation . . . . . . . . . . . . . 91

5.1.2 Perpendicular Alignment of Force and Oscillation . . . . . . . . . . 92

5.1.3 Cylindrical Roller and Parallel Alignment . . . . . . . . . . . . . . 93

5.1.4 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.5 Analysis of Deviations . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Experimental Rig for the Micro-Walking Effect . . . . . . . . . . . . . . . 103

5.2.1 Parameters of the Experimental Rig . . . . . . . . . . . . . . . . . 104

5.2.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . 108

5.2.3 Analysis of Deviations . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Conclusion and Outlook 115

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1.1 Shakedown and Ratcheting in Incomplete Contacts . . . . . . . . . 115

6.1.2 Dynamic Influences on Sliding Friction . . . . . . . . . . . . . . . . 117

6.1.3 The Interplay of System Dynamics and Friction . . . . . . . . . . . 118

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2.1 Shakedown and Ratcheting in Incomplete Contacts . . . . . . . . . 119

6.2.2 Dynamic Influences on Sliding Friction . . . . . . . . . . . . . . . . 120

References 120

vi

A Statistics 130

A.1 Analysis of the Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.1.1 Specification of Measurement Results . . . . . . . . . . . . . . . . 130A.1.2 Identification of Parameters . . . . . . . . . . . . . . . . . . . . . . 131A.1.3 Curve Fitting and Regression . . . . . . . . . . . . . . . . . . . . . 131A.1.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.2 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B Numerical Methods 135

B.1 Incremental MDR Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 135B.2 CONTACT Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.3 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139B.4 Contact Models for the Micro-Walking Machine . . . . . . . . . . . . . . . 139

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viii

List of Figures

1.1 frictional contact of two bodies . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 micro-slip and rigid body motion . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 sketch of the influence of system dynamics on systems with friction . . . . 7

2.1 complete and incomplete contacts . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 indentation of an elastic half-space by a rigid sphere . . . . . . . . . . . . 11

2.3 spherical tangential contact . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 rolling contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 first step of the MDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 second step of the MDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 tangential contact with MDR . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 rolling contact with MDR . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 velocity dependent coefficient of friction . . . . . . . . . . . . . . . . . . . 25

2.10 contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 oscillating rolling contact and time response of the rolling . . . . . . . . . 28

3.2 oscillating rolling leads to varying reduction of the friction bound . . . . . 29

3.3 displacement of the substrate in case of shakedown . . . . . . . . . . . . . 30

3.4 displacement of the substrate in case of ratcheting . . . . . . . . . . . . . 30

3.5 MDR model of the tangential contact . . . . . . . . . . . . . . . . . . . . 31

3.6 alternating contact and displacement of the pressure distribution . . . . . 32

3.7 distribution of stick and slip in case of shakedown . . . . . . . . . . . . . . 32

3.8 normalized dissipation per time step for different relative accuracies . . . 33

3.9 normalized rigid body shift per time step for different relative accuracies . 33

3.10 traction after shakedown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.11 traction after shakedown at the centre line . . . . . . . . . . . . . . . . . . 34

3.12 tangential displacement after shakedown . . . . . . . . . . . . . . . . . . . 35

3.13 tangential displacement after shakedown at the centre line . . . . . . . . . 35

3.14 displacement before and after shakedown . . . . . . . . . . . . . . . . . . 37

3.15 displacement after shakedown . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.16 shakedown displacement as a function of fT for different w . . . . . . . . 38

3.17 maximal amplitude wlim as a function of the tangential force . . . . . . . 39

3.18 maximal displacement ulim as a function of the tangential force . . . . . . 39

3.19 distribution of stick and slip in case of ratcheting . . . . . . . . . . . . . . 40

ix

3.20 coefficient η as a function of the static displacement . . . . . . . . . . . . 41

3.21 incremental displacement ∆u as a function of the tangential force . . . . . 41

3.22 static contact and perpendicular oscillating rolling contact . . . . . . . . . 42

3.23 shakedown for parallel and perpendicular setting . . . . . . . . . . . . . . 42

3.24 ratcheting for parallel and perpendicular setting . . . . . . . . . . . . . . . 42

3.25 traction after shakedown along the parallel centre line . . . . . . . . . . . 43

3.26 traction after shakedown along the perpendicular centre line . . . . . . . . 43

3.27 displacement after shakedown along the parallel centre line . . . . . . . . 44

3.28 displacement along the perpendicular centre line . . . . . . . . . . . . . . 44

3.29 one-dimensional MDR model for parallel and perpendicular setting . . . . 45

3.30 shakedown displacement usd as a function of fT for different w . . . . . . 46

3.31 maximal amplitude as a function of the maximal tangential force . . . . . 47

3.32 maximal displacement as a function of the maximal tangential force . . . 47

3.33 incremental displacement ∆u for w ⊥ fT . . . . . . . . . . . . . . . . . . . 48

3.34 tangentially loaded contact of a rigid cylinder and an elastic half-space . . 49

3.35 oscillating rolling contact of rigid cylinder and elastic half-space . . . . . . 52

3.36 displacement u for different oscillation amplitudes w and fT = 0.08 . . . . 52

3.37 displacement u for different oscillation amplitudes w and fT = 0.58 . . . . 52

3.38 pressure in the reversal points and traction after shakedown . . . . . . . . 53

3.39 tangential displacement of particles ux for different fT . . . . . . . . . . . 55

3.40 normalized half stick width csd after shakedown . . . . . . . . . . . . . . . 55

3.41 normalized tangential traction after shakedown for different fT . . . . . . 56

3.42 normalized tangential traction after shakedown for different w . . . . . . . 56

3.43 shakedown displacement usd as a function of the tangential force fT . . . 57

3.44 maximal amplitude wlim as a function of the tangential force . . . . . . . 58

3.45 maximal displacement ulim as a function of the tangential force . . . . . . 58

3.46 incremental displacement ∆u as a function of the tangential force . . . . . 59

4.1 basic model for the micro-walking effect . . . . . . . . . . . . . . . . . . . 61

4.2 elastic specimen and simplified rigid body model . . . . . . . . . . . . . . 64

4.3 elastic contact spot and spring element . . . . . . . . . . . . . . . . . . . . 65

4.4 free body diagram of the non-dimensional system . . . . . . . . . . . . . . 67

4.5 steady state motion for parameter set A . . . . . . . . . . . . . . . . . . . 68

4.6 steady state motion for parameter set B . . . . . . . . . . . . . . . . . . . 68

4.7 main effect-plot for the effective friction . . . . . . . . . . . . . . . . . . . 71

4.8 main effect-plot for the minimal amplitude . . . . . . . . . . . . . . . . . . 73

4.9 parameter study of κ8 and µ for µe . . . . . . . . . . . . . . . . . . . . . . 74

4.10 parameter study of κ8 and µ for zmin/zstat . . . . . . . . . . . . . . . . . . 74

4.11 free body diagram of the system in a tilted position . . . . . . . . . . . . 75

4.12 parameter study of κ8 and µ for ts1 . . . . . . . . . . . . . . . . . . . . . . 77

4.13 parameter study of κ8 and µ for ts2 . . . . . . . . . . . . . . . . . . . . . . 77

4.14 parameter study of κ8 and µ for r1 . . . . . . . . . . . . . . . . . . . . . . 78

4.15 parameter study of κ8 and µ for r2 . . . . . . . . . . . . . . . . . . . . . . 78

x

4.16 main effect-plot for the minimal amplitude . . . . . . . . . . . . . . . . . . 79

4.17 phase space diagram of the lateral motion . . . . . . . . . . . . . . . . . . 80

4.18 motion of the system for the maximal reduction range . . . . . . . . . . . 80

4.19 phases of motion of the specimen in the vanishing reduction range . . . . 80

4.20 effective coefficient of friction µe for varying base velocity . . . . . . . . . 83

4.21 time course of the spring force for set 1 and set 2 . . . . . . . . . . . . . . 84

4.22 relative time of instability ti/T as a function of the base velocity . . . . . 84

4.23 specimen with different positions of the lever arm . . . . . . . . . . . . . . 85

4.24 surface plot of µe as a function of v0 and κ8 . . . . . . . . . . . . . . . . . 86

4.25 area plot of ti/T as a function of base velocity v0 for different κ8 . . . . . 86

4.26 effective coefficient of friction µe as a function of base velocity v0 . . . . . 87

5.1 rolling bodies experimental test rig . . . . . . . . . . . . . . . . . . . . . . 90

5.2 experimental rig for the oscillating rolling with measurement chain . . . . 90

5.3 rolling body experimental rig with sphere and parallel set-up . . . . . . . 91

5.4 rolling body experimental rig with sphere and perpendicular set-up . . . . 92

5.5 rolling body experimental rig with cylindrical roller and parallel set-up . . 93

5.6 displacement u for different oscillation amplitudes w . . . . . . . . . . . . 95

5.7 displacement u for fT = 0.64 and stepwise increase of w . . . . . . . . . . 95

5.8 displacement u for fT = 0.42 and stepwise increase of w . . . . . . . . . . 96

5.9 conditioned displacement u and polynomial fit . . . . . . . . . . . . . . . 96

5.10 displacement u for different oscillation amplitudes w . . . . . . . . . . . . 99

5.11 height difference between bearing and contact point . . . . . . . . . . . . 99

5.12 model of the contact between roller and elastic layer of thickness b . . . . 100

5.13 optical measurement of the contact area . . . . . . . . . . . . . . . . . . . 100

5.14 boundary effects of the cylindrical roller . . . . . . . . . . . . . . . . . . . 101

5.15 contact of rubber substrate and standard model . . . . . . . . . . . . . . . 102

5.16 experimental rig for the micro-walking effect with measurement chain . . 103

5.17 lateral and front view of the hourglass-shaped specimen . . . . . . . . . . 104

5.18 experimental rig for the micro-walking effect . . . . . . . . . . . . . . . . . 104

5.19 contact of elastic body and rigid substrate with contact diameter Dc . . . 106

5.20 time response of Fs-smooth sliding . . . . . . . . . . . . . . . . . . . . . . 109

5.21 time response of Fs-self-excited oscillations . . . . . . . . . . . . . . . . . 109

5.22 close up of the time response of Fs-smooth sliding . . . . . . . . . . . . . 109

5.23 close up of the time response of Fs-self-excited oscillations . . . . . . . . . 109

5.24 time response of the spring force Fs for the parameter set 5 . . . . . . . . 111

5.25 frequency spectrum Ss−exp of Fs for the for the parameter set 5 . . . . . . 111

5.26 characteristic frequency Ωexp as a function of the velocity v0 . . . . . . . . 112

5.27 characteristic frequency ωsim as a function of the velocity v0 . . . . . . . . 112

5.28 time response of the spring force for experiment and simulation . . . . . . 113

5.29 beam model of the connection element of the system . . . . . . . . . . . . 114

6.1 scenarios in mechanical systems with friction and oscillations . . . . . . . 118

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A.1 design of experiments approach . . . . . . . . . . . . . . . . . . . . . . . . 134

B.1 stepwise incremental simulation scheme . . . . . . . . . . . . . . . . . . . 136B.2 numerical model of the CONTACT software . . . . . . . . . . . . . . . . . 137B.3 2D and 3D contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

xii

List of Tables

4.1.1 definition and interpretation of the remaining parameters . . . . . . . . . 664.2.1 parameters that are constant in the detailed parameter study . . . . . . . 744.3.1 parameters of the different sets used in the experiment . . . . . . . . . . . 834.3.2 maximum parameter range of the experimental set . . . . . . . . . . . . . 86

5.1.1 parameters of the experimental rig with sphere and parallel alignment . . 925.1.2 parameters of the experimental rig with sphere and perpendicular set-up . 935.1.3 parameters of the experimental rig with cylinder and parallel set-up . . . 945.2.1 parameters of the micro-walking experimental rig . . . . . . . . . . . . . . 105

A.1.1quality of the curve fitting results . . . . . . . . . . . . . . . . . . . . . . . 132

xiii

NomenclatureParameters with Latin Letters

symbol unit parameter

a m contact radiusai - factor value i in DoE methodAs m2 cross sectional area of spring rodAc m2 contact area

Aslip m2 slip areab m slip radiusbi - factor value i in DoE methodc m stick radiuscs m s−1 speed of soundcsd m stick radius after shakedownd m indentation depthD m contact disc diameter of micro-walking machineDb m bulk body diameter of micro-walking machineDc m contact lengthE N m−2 Young’s modulusE∗ N m−2 effective Young’s modulus

f (xi, αj) 1 model function for least square methodfM 1 friction parameter in the model of Martins et al.fs 1 normalized spring force of base spring

fsamp Hz sampling frequency of micro-walking machine test rigfT 1 normalized tangential force

fT,lim 1 normalized maximal forcefx N tangential spring force of elastic foundationfz N normal spring force of elastic foundation

fs 1 mean of normalized spring force of base spring

fs 1 normalized spring force of base spring

fx N test tangential spring force of elastic foundationf0 Hz excitation frequency of rolling body test rigf1 Hz first eigenfrequency of rolling body test rig

FbZ N elastic force of beam model of the springFcZ N contact force of beam model of the springFN N normal force

FN1/2 N normal force of contact springs

FR N resistance forceFs N spring force of base springFT N tangential force

xiv


FT,max N maximum holding forceFT 1/2 N tangential force of contact springs

FW N resistance force of cross roller table

FNi N average mean normal forceg m s−2 gravitational acceleration

g (x) m equivalent profile functionG - function for determination of measurement uncertaintyG N m−2 shear modulusG∗ N m−2 effective shear modulusG′ N m−2 storage modulus of standard rubber modelG′′ N m−2 loss modulus of standard rubber modelh m lever arm of base spring

h (r) m profile functionhM 1 ratio of height and width in the model of Martins et al.

hk (x) m actual profile functionh0 (x) m initial profile function∆h m height difference between bearing and contact pointIs m4 geometrical moment of inertiak m constant of proportionalityks N m−1 spring stiffness of base springkx N m−1 tangential spring stiffness of contact spotskz N m−1 normal spring stiffness of contact spots

ks N m−1 effective spring stiffness of base spring∆kx N m−1 tangential spring stiffness of elastic foundation∆kz N m−1 normal spring stiffness of elastic foundationls m length of spring rod of micro-walking machinel1 m length of centre part of micro-walking machinel2 m length of side disc of micro-walking machineLc m contact widthm kg mass of the micro-walking machine

mT kg mass of weight pot and weightMbΦ N m elastic moment of beam model of the springMcΦ N m moment of contact of beam model of the springp (x) N m−2 pressure distributionp0 N m−2 maximal pressure

p01−12 1 coefficients of polynomial fit functionp′

0 N m−2 reduced pressure at boundary of cylindrical contactr m coordinate in polar-coordinate systemr 1 Bravais-Pearson correlation coefficientrs m radius of cross sectional area of spring rodr0 m constant radius in 2-D displacement function

xv


R m radius of sphere and cylinderRc m radius of contact disc of micro-walking machine

R1D m radius of stretched 1-D profiles m coordinate of spring position in MDR model

sM 1 stiffness parameter in the model of Martins et al.s1/2 1 normalized motion of contact spots

s′1/2 1 normalized velocity of contact spots

Sj - sum of squared deviations for set j of model parametersSs 1 frequency spectrum of normalized spring forcet 1 dimensionless timet 1 correction term of student-t distribution

tdim s dimensional timetis 1 relative stick time of instability

ts1/2 1 relative stick time of contact spots

t0 1 point in time after the transient processt∗ 1 shifted timet 1 conditioned and normalized timeT s, 1 time of observationTc s characteristic time of loading

Ts1/2 s stick time of contact spots

u 1 normalized displacementun1/2 1 normalized normal deflection of contact springs

ulim 1 normalized maximal rigid body displacementun1/2 1 normalized tangential deflection of contact springs

usd 1 normalized shakedown displacementustat 1 normalized static displacement

ux (x) 1 normalized tangential displacement of particlesux1/2 1 normalized tangential deflection of contact springs

uz1/2 1 normalized normal deflection of contact springs

u 1 conditioned and normalized displacementux1/2 1 normalized test tangential deflection of contact springs

us1/2 1 normalized test tangential deflection of contact springs

uz1/2 1 normalized test normal deflection of contact springs

uz1/2 1 normalized normal deflection of contact springs

∆u 1 normalized incremental displacementU m relative displacement of particles

Umax m maximal static displacement of spherical contactU0 m maximal static displacement of cylindrical contactUrt m ratcheting displacementUsd m shakedown displacement

Ustat m static stick region displacement

xvi


Ux m tangential displacement of particlesUx (x) m tangential displacement of springs of elastic foundationUx1/2 m tangential deflection of contact springs

Ux (x) m test tangential displacement of springs of elastic foundationUz m vertical displacement of particles

Uz (x) m vertical displacement of springs of elastic foundationUz1/2 m normal deflection of contact springs

∆U m incremental displacement∆Us m rigid body shift per time step

∆U m displacement caused by height difference

U m s−1 slip velocity of particlesv m s−1 velocity

var (fs) m s−1 revolving variance of spring forcev0 m s−1 velocity of base point of micro-walking machine∆v m s−1 limiting velocity in µ-function∆v1 m s−1 velocity difference of MDR model of rolling contactw 1 normalized rolling amplitude

wlim 1 normalized maximal rolling amplitudeW m rolling amplitude

Wfric N m frictional dissipation per time stepW0 N m frictional dissipation per cycle of tangential loading∆W m incremental increase of rolling amplitude

x m coordinate in Cartesian coordinate systemx 1 normalized lateral coordinate of translational motionxi - specific value of a measurement ixs 1 normalized lateral coordinate of base pointx′ 1 normalized lateral velocity of micro-walking machinex′

s 1 normalized lateral velocity of base pointx′′ 1 normalized lateral acceleration of translational motionx - arithmetic mean value of measurements

∆x m grid-size of MDR and CONTACT models∆x 1 normalized difference of lateral coordinate of translational motion∆x 1 normalized difference of lateral coordinate of translational motionX m lateral coordinate of translational motion

X m s−2 lateral acceleration of translational motiony m coordinate in Cartesian coordinate systemyi - specific value of a measurement variable iy - arithmetic mean value of measurementsyi - fit value of a measurement variable iδyi - residual of a measurement variable i∆y m grid-size of CONTACT models

xvii


z m coordinate in Cartesian coordinate systemz 1 normalized vertical coordinate of translational motion

zstat 1 normalized static vertical displacementz′′ 1 normalized vertical acceleration of translational motionz 1 normalized vertical coordinate of translational motionZ m vertical coordinate of translational motion

Z m s−2 vertical acceleration of translational motion2a m height of micro-walking machine2b m width of micro-walking machine

2tD s size of Dirichlet window for revolving variance

Parameters with Greek Letters


αj - set j of model parametersβ 1 Dundurs’ constantγ s m−1 steepness parameter in µ-functionε 1 relative accuracy of output variables

εxx 1 tangential strainζ m moving coordinateη m coordinate in Cartesian coordinate systemη Pa s viscosity of standard rubber modelη 1 constant of proportionality of incremental displacement

ηexp 1 constant of proportionality-experimentηfit 1 constant of proportionality-fit

ηMDR 1 constant of proportionality-MDRΘ kg m2 rotational inertia of micro-walking machineκ 1 mapping parameter

κ1−8 1 non-dimensional parameters of micro-walking machineµ 1 coefficient of frictionµc 1 critical coefficient of frictionµe 1 effective coefficient of frictionµk 1 kinematic coefficient of frictionµs 1 static coefficient of frictionν 1 Poisson’s ratioξ m coordinate in Cartesian coordinate systemξx 1 creep ratio of 3-D rolling contactξ1 1 creep ratio of 1-D rolling contact

xviii


σx - standard deviation of the mean valueσy - standard deviation of the mean valueτ s characteristic period of micro-walking machine

τ (x) N m−2 traction distributionτmax N m−2 traction bound

τr s relaxation timeτR N m−2 resistance traction

τ1 (x) N m−2 static traction distributionτ1−sd (x) N m−2 shakedown traction distribution

τ2 (x) N m−2 additional static traction distributionτ2−sd (x) N m−2 shakedown traction distribution

τ2 N m−2 correctional traction termτ21 N m−2 correctional shakedown traction termτ22 N m−2 correctional shakedown traction termϕ 1 normalized rigid body rotation angleϕ′′ 1 normalized angular acceleration of rigid bodyϕ 1 normalized rigid body rotation angleΦ 1 rotation angle of rigid body

Φ 1/s2 angular acceleration of rigid bodyω 1/s angular velocity of rolling sphere

ωsim 1 normalized characteristic frequency of simulationΩexp 1/s characteristic frequency of experimentΩsim 1/s characteristic frequency of simulation

xix

xx

Chapter 1

Introduction

Frictional resistance is a crucial parameter in various technical, seismological and evenbiological systems. The core problem can be broken down to the frictional interaction oftwo bodies as depicted in Fig. 1.1 (a) that is often modelled using the so called Coulomb’slaw of dry friction. This model goes back to the works of Amontons [1] and Coulomb [2]and gives a proportional relation between the normal load FN and the tangential loadFT of the contact.

xy

z

FN

FTµs, µk

(a) macroscopic system

p (x, y)τ (x, y)

xy

z

FN

FT

(b) microscopic contacts

Fig. 1.1: frictional contact of two bodies loaded in the normal and tangential directionwith FN and FT . (a) macroscopic system. (b) free body diagram with microscopic contacts

However, one distinguishes two cases. If FT falls below the maximum holding forceFT,max the system remains at rest:

FT ≤ FT,max = µsFN . (1.0.1)

This refers to the state of stick. Otherwise a relative movement of the two bodiesoccurs, where each of the two is subjected to the resistance force FR which is caused bydissipative processes in the contact:

FR = µkFN . (1.0.2)

This refers to the state of slip, where the resistance force acts in the opposite directionof the slip direction. The two constants of proportionality denote the so called static(µs) and dynamic (µk) coefficient of friction. One explanation for the proportionality

1

Chapter 1. Introduction

of Coulomb’s law lies in the contact properties of rough surfaces. According to thetheory by Bowden and Tabor [3] rough surfaces consist of numerous small hills namedasperities. Thus, the two bodies only touch in a few contact spots where, in case of drymetals, cold-weld junctions are formed. Assuming that the number of contact spots isproportional to the normal load, the coefficient of friction equals the ratio of the tractionnecessary to shear the cold-weld junctions and the penetration hardness. This explainsalso why experimentally, the static coefficient µs often exceeds the kinetic one µk sincethe metallic junctions become stronger after the surfaces have been in static contact forsome time [4]. The coefficients are determined experimentally and are classified for alarge number of combinations of contacting materials. This often leads to the widespreadmisconception of Coulomb’s law of dry friction according to which the coefficients offriction are true constants that only depend on intrinsic properties of the materials incontact [5]. In fact they depend on small scale effects in the contact and are influencedby a various time and environmental conditions for both lubricated and dry contacts1

[3]. In particular, the material combination, the contact geometry, chemical reactions,the temperature and the normal force itself all play an important role [5]. The staticcoefficient is also affected by the history of the contact, as evidenced by the fact thatit increases with time [4, 7]. Furthermore, the kinetic coefficient is strongly affected bydynamic influences as sliding velocity [8] and vibrations [9]. This is a particular problemas it affects every measurement apparatus, i.e. tribometer, that is used for determinationof the coefficients. On the one hand, this limits the value of tabulated coefficients asthe influence of the tribometer used in the experiment remains vague. On the otherhand, this is a possible explanation for difficulties on the reproducibility of measurementof coefficients that are determined with different tribometers under otherwise similarconditions [10].In order to illustrate the basic underlying mechanism that is responsible for the influenceof the systems motion, i.e. the system dynamics, one can consider a microscopic modelof dry friction with only one constant coefficient µ. This coefficient is intended to reflectall microscopic influences on the surface that depend on the material pairing. Regardlesswhether the system consists of one single contact or numerous contact zones as in roughsurfaces, the contact loads will occur as a stress distribution on the real contact areaAc. One distinguishes the pressure p (x, y), which is the stress that acts perpendicularto the tangent of the contact surface, and the traction τ (x, y), which acts parallel to thetangent, as shown in Fig. 1.1 (b). Thus, the microscopic equivalent to Coulomb’s lawstates that a particle at the contact surface at position (x, y) is at rest as long as:

τ (x, y) ≤ τmax = µp (x, y) , (1.0.3)

and starts to slip as soon as this so called traction bound, i.e. the local maximal tan-gential load, is exceeded. In this case, the particle is subjected to a resistance traction:

τR (x, y) = µp (x, y) . (1.0.4)

1It should be emphasized that Coulomb was already fully aware of these many influences and alsopresented a lot of experimental work on this subject as shown in the article of Popova and Popov [6].

2


For real static systems or stationary dynamic systems with constant relative velocitybetween the two contacting bodies, the microscopic criteria (1.0.3) and (1.0.4) wouldnot imply any difference to the macroscopic criteria (1.0.1) and (1.0.2), except for theconstant µ. However, the case is different if one considers a pressure distribution p (x, y, t)that varies in time. The particles on the surface will start to slip whenever the tractionbound (1.0.3) is exceeded. Assuming that adjacent contact zones are sufficiently faraway from one another they could be regarded as being independent. This gives thepossibility that some of them are in a state of stick (stick zones) while others are in astate of slip (slip zones). This applies even under the assumption that the macroscopictangential force does not exceed the maximal tangential load at any time:

FT (t) =

∫

Ac

τ (t)dA < µFN (t) ∀t . (1.0.5)

In case that slip occurs successively in all the contact zones it would consequently accu-mulate to a continued rigid body motion even if assumption (1.0.5) is never violated. Forinstance, one can consider an elastic substrate and a rigid block that is constantly loadedas in Fig. 1.2 (a). A slight oscillatory rotation will lead to a time varying pressure suchthat slip alternately occurs on one side while the other side sticks as shown in Fig. 1.2(b). Thus within every period the system will move an incremental step ∆U to the rightas shown in Fig. 1.2 (c). The micro-slip induces a rigid body motion. However, thisscenario also requires that the slip is always directed in the same direction, otherwise nonet rigid body motion would occur.

FN

FT

(a) rigid block

stickstick slipslip

(b) alternating slip

∆U

(c) motion

Fig. 1.2: (a) constantly loaded rigid block on elastic substrate. (b) oscillatory rotationleads to varying slip and stick zones. (c) micro-slip causes rigid body motion ∆U

Considering mechanical systems with friction this introductory example raises severalissues. Firstly, micro-slip can lead to failure of nominally static systems. By this,one means the effect that micro-slip can cause relative motion of technical componentsthat should actually not move according to the macroscopic criterion (1.0.1). Secondly,micro-slip can explain the dynamic influence on the frictional resistance of systems withgross slip. This describes the effect that a macroscopic motion that appears as smoothsliding can in fact be superposed by microscopic vibrations in the contact zone. Thesevibrations might reduce the frictional resistance of the overall system such that it isactually lower than estimated considering the macroscopic criterion (1.0.2).

3


1.1 State of Scientific Research

In the following section some important works will be introduced that serve as thestarting point for the following analysis. A more or less arbitrary classification intonominally static and dynamic systems is chosen, although such a strong distinction doesnot exist. Here, nominally static refers to systems in which no gross sliding and thusno relative motion of components is initially expected. Dynamic refers to the opposite,those systems in which gross sliding and relative motion is expected to occur.

1.1.1 Nominally Static Systems

Force locked connections are found in various technical systems. These nominally staticfrictional contacts2 are crucial for the generation of solid detachable and non-detachableconnections between technical components. Examples include bolted connections [11,12], machining fixtures [13], interference fits [14] or dovetail connections on the hub offan-blades [15]. The tangential load capacity of these systems is primarily determined bythe macroscopic normal load FN and the coefficient of friction µ. According to Coulomb’slaw of dry friction such a connection holds as long as the macroscopic tangential load FT

falls below the maximum holding force FT,max of equation (1.0.1). Due to vibrations, themacroscopic loads often either consist of a significant static part superposed by smalloscillations, or the overall loads are constant and there occur only oscillations of thestress distribution. For example, in the dovetail connections of turbine blades the radialcentrifugal force pulls the dovetail into the socket while there is a superimposed vibrationin the plane of the fan. These scenarios can lead to a periodic incremental slip U (x, y) ofthe particles in the interface, even if FT is far below FT,max, i.e. appears insufficient tocause complete sliding [16]. Possible consequences of this are micro-slip [17] and frettingfatigue [18, 19] of the relevant components.

Frictional Shakedown Churchmann and Hills [20] and Antoni et al. [21] showedthat under certain conditions, it happens that the interface slip ceases after the first fewoscillation periods. This is the case, if the initial displacement of particles generates aresidual force in the interface, which is sufficiently strong to prevent any further slip.Subsequently, the entire contact will finally remain in a state of stick even if the oscillationis continued. Due to the strong analogy to the shakedown state in solid mechanics, inwhich the deformed bodies only show plastic strain in the first few loading cycles andpure elastic response afterwards [22], this effect is referred to as frictional shakedown[23]. Consequently, Klarbring et al. transferred the well-known Melan theorem forplastic shakedown [24] to both discrete [25] and continuous systems [26] with Coulombfriction and incomplete contact, i.e. systems in which the contact area does not changeduring the loading cycle. The shakedown theorem states that if there exists a safeshakedown displacement of particles in the interface U (x, y) for which the whole systemis in a state of stick, then the actual slip displacement U (x, y) will monotonically reach

2This refers to connections that are intended to hold.

4


U (x, y) for all (x, y) ∈ Ac. Thus, a system that is subjected to an oscillating load, willshakedown and monotonically reach a safe shakedown displacement, i.e. an equilibriumstate without any slip and dissipation, even if the oscillation is continued. One importantrequirement for this is an uncoupled system, meaning that tangential displacements inthe interface do not influence the pressure and vice versa. However, in case of coupledtwo dimensional discrete systems, shakedown is also possible, if the friction coefficient ineach node is less than a critical value, which depends on the coupling between adjacentnodes [25].

Ratcheting In case that the system exceeds the shakedown limits, i.e. the prereq-uisites for a safe shakedown to occur, the micro-slip continues and the system exhibitssteady state slip. One possible scenario is that the micro-slip occurs in different con-tact areas for each loading cycle. In turn, the accumulation over all the cycles leadsto a rigid body motion. This case is referred to as ratcheting due to the similarity toplasticity, where ratcheting specifies an accumulating deformation that usually leads toplastic collapse of the system [22, 23]. Antoni et al. stated that ratcheting can leadto significant global relative displacement between components and can account for thefailure of some assembly parts in mechanical structures [21]. Mugadu et al. introduceda rigid flat punch loaded by a constant tangential force insufficient to cause gross sliding[27]. A constant normal force that moves backwards and forwards on the punch causesmicro-slip that alternately migrates in from one side of the contact. This induces arocking motion of the punch such that it walks across an elastic half plane by a constantincrement in each loading cycle, even though the global friction limit is not exceeded atany time. Nowell used numerical methods to also study the case for a normal force beingstrong enough to completely lift the punch from the ends [28]. Ciavarella extended themodel taking into account dissimilar materials and showed that dissimilarity decreasesthe load needed for ratcheting to occur and increases the slip per cycle [29].

Cyclic Slip In addition, it is possible, that a mechanical system with friction exhibitsreversing slip with no net rigid body displacement. This case is referred to as cyclic slipin accordance to the cyclic plasticity case [23]. Ciavarella showed that for the walkingpunch model cyclic slip only occurs for dissimilar materials, as for similar materials, aconstant tangential load leads to a constant direction of slip [29].

1.1.2 Dynamic Systems

Many systems also exhibit sliding friction [30]. This describes the situation when twocontacting bodies move relatively to each other. Examples include brakes [31], clutches[32], machine tool slide-ways [33], joints in technical [34] and biological systems [35]and even seismological systems as tectonic plates [36]. Generally spoken, sliding frictionoccurs whenever FT exceeds the maximum holding force. The system is then subjectedto the resistance force FR as in equation (1.0.2) that is caused by dissipative processes inthe contact. Despite the fact that the kinetic coefficient of friction µk can be measured

5


with little difficulty under laboratory conditions, there exist severe dynamic influences.Firstly, this makes it very difficult to predict the coefficients a priori from first principles.Secondly, this limits the value of tabulated coefficients as the influence of the tribometerused in the experiment remains unclear [5, 37].

System Dynamics Dynamic influences on the frictional resistance include for instancedamping, self-excited and externally excited vibrations, the velocity, the contact stiffnessand the interaction between the structural dynamic behaviour and the excitation by thefrictional force [5]. In particular the impact of system dynamics on the interface responseof frictional contacts is a crucial factor. This is mainly caused by a coupling between thein-plane and out of plane vibrations in sliding systems [38]. An important work is theexperiment of Tolstoi, who studied the interaction of the tangential and normal degreeof freedom [39]. He found that tangential slip events are accompanied by an upwardmovement of the contacting body and that damping increases the frictional resistance.In addition, he used piezo actuators to induce externally excited vibrations in the normaldirection. Through resonance effects, the frictional resistance was reduced between 35 %and 85 %. Godfrey used electrical resistance measurements and showed that this effectis due to a reduction of the metal-to-metal contact zone induced by decreasing loads.The reduction in the coefficient of friction was therefore characterized as apparent [40].Polycarpou and Soom used linear dynamic models to compute the normal motion ofa lubricated sliding system. They concluded that the instantaneous normal separationsignificantly affects the friction force. A good representation of the dynamics of thesliding system in the normal direction is therefore very important [41, 42, 43]. Severalauthors added a rotational degree of freedom. By this, they considered that the kine-matic coupling between normal, tangential and rotational motion leads to varying forcesand moments what in turn influences the friction [10, 44, 45]. Twozydlo et al. modelleda typical pin-on-disk apparatus consisting of rigid bodies with elastic connections. Theyshowed that coupling between rotational and normal modes induces self-excited oscil-lations. In combination with high-frequency stick-slip motion, these oscillations reducethe apparent kinetic coefficient of friction. A particular pin-on-disk experimental set-upgave good qualitative and quantitative correlation with numerical results [46]. Adamsexamined a beam that is in frictional contact with an elastic foundation [47]. In thismodel the interplay of the friction force and moments at the interface and the bending ofthe beam leads to instabilities that increase with increasing coefficient of friction. Thisself-excited oscillation leads to a partial loss of contact and a stick-slip motion.

Slip Waves Despite discrete lumped parameter models, many authors considered in-terface waves with separation between contacting half-spaces. Caminou and Dundursinvestigated the so-called carpet-fold motion, which gives the possibility of a sliding mo-tion between two identical bodies without interface slipping [48], i.e. without frictionalresistance. Adams examined elastic half-spaces in dry contact and concluded that thedeformation along the interface and slip wave propagation is a potential destabilizingmechanism for steady sliding [49]. Although his model uses a constant coefficient of

6


friction without distinction between the static and kinetic case, he demonstrated thatinterface slip waves may be responsible for the apparent velocity-dependence of frictioncoefficient measurements [50]. The apparent coefficient of friction can decrease withincreasing sliding speed due to the carpet-fold motion.

1.2 Objective

On basis of the studies described above, theoretical models and experiments are to bedeveloped in order to examine the influence of micro-slip on the macroscopic behaviourof mechanical systems with friction. As sketched in Fig. 1.3 system dynamics and micro-slip affects nominally static systems as well as dynamic systems.

system dynamics

nominallystatic systems

dynamicsystems

failure ofconnections

frictionalresistance

Fig. 1.3: sketch of the influence of system dynamics on systems with friction

Starting from the distinction introduced in section 1.1 the objective of this work isstructured as follows.

Shakedown and Ratcheting Shakedown characterizes the effect that oscillatory fric-tional slip causes a state of residual stress that inhibits or reduces the slip in subsequentcycles. It is worthwhile to predict under which conditions a system responses like thisbecause shakedown is one way to prevent for instance fretting or failure of force lockedconnections. So far the theory of frictional shakedown was examined only for completecontacts but much less is known for the case where the contact area changes with time[29]. In addition, many studies are based on numerical simulations and the results areoften difficult to interpret and cannot easily been transferred to slightly modified systems[28]. Thus, the objective is to introduce a frictional system with an incomplete contactand to study the different scenarios that occur if this system is subjected to oscillatingloads. More specifically the first objectives for this work are as follows:

• Development of generic models for incomplete contacts

• Analysis of the shakedown process and the final shakedown state

• Derivation of the shakedown limits, i.e. failure criteria

• Examination of the ratcheting phenomenon in incomplete contacts

7


Dynamic Influences on Sliding Friction Vibrations in the normal direction de-crease the frictional resistance of mechanical systems. This effect has been known forlong and is also applied in technical systems [39, 51]. Furthermore friction and cer-tain dynamic coupling effects can lead to instabilities [49, 50]. This raises the questionto what extent these self-excited vibrations can lead to a self-excited reduction of themacroscopic frictional resistance. In case that the corresponding amplitudes are of amicroscopic character, they could be superimposed to an apparently smooth sliding mo-tion of the system while being undetected. Hence, it is one aim of this work to developa model that exhibits this kind of behaviour. More specifically, the further objectivesare as follows:

• Introduction of a model that captures dynamic influences on sliding friction

• Investigation of the post-instability behaviour and analysis of the parameters

• Analysis of the underlying basic mechanisms for the reduction

In order to achieve these aims, classical contact mechanics, the well-known Methodof Dimensionality Reduction, several numerical simulation tools and experiments arebeing used. The final goal is not only to gain a deeper understanding of the effectsof shakedown, ratcheting and walking, but also to give guidelines for the constructionof real technical systems. Examples range from force-locked connections to mechanicaldrives and to tribometers.

1.3 Structure

Methods and theories that are important for the further analysis are introduced inchapter 2. In addition to a short description of important contact systems, the Methodof Dimensionality Reduction is described and different aspects of the modelling of dryfriction are discussed. This gives the fundamentals for the next steps of the analysis.Firstly, chapter 3 focuses on shakedown and ratcheting in incomplete contacts. Severalmodels are introduced and analysed and the shakedown limits are derived. Secondly,in chapter 4, the phenomenon of self-induced oscillations and the resulting reduction ofsliding friction is discussed. Both topics are followed by an experimental investigation.The experimental rigs that are used in the analysis and the experimental procedures areintroduced in chapter 5. Finally a conclusion and an outlook are given in chapter 6. Inthe appendix, for reasons of reproducibility, methods and theories are described that areused in this work but do not contribute substantially to the results.

8

Chapter 2

Fundamentals and Methods

In general, mechanical systems with friction are influenced by a complex interaction ofdifferent phenomena:

Rigid Body Motion The relative motion of the contacting bodies influences thestresses in the contact. The motion has its origin in the forces and moments acting onthe bodies but is also influenced by inertia effects.

Contact Mechanics The stresses in the contact lead to deformation of the materialsof the contacting bodies and thus change the contact area. This deformation dependson the geometry of the bodies as well as their material behaviour.

Friction The distribution of stick and slip in the contact area is a result of the contactstresses. However, this in turn affects the forces acting on the bodies.

These many cross influences complicate the understanding and the analysis of the in-terplay of system dynamics and friction. In the following section, the basic tools andmodels used in this work are introduced. In addition to a brief summary of importantcontact systems, the Method of Dimensionality Reduction (MDR) is introduced. Fur-thermore, some important aspects in the modelling of frictional systems are discussed.This enables an accurate and efficient modelling of mechanical systems with frictionunder consideration of dynamic influences.

2.1 Contact Mechanics

In case that the contact area and the slope of the geometry of the bodies, respectivelytheir deformation, are small in comparison to the size of the contacting bodies, they maybe represented by half-spaces [52]. This assumption describes a semi-infinite body whoseonly boundary is an infinite plane what greatly simplifies the calculation of elastic contactproblems. In general, one distinguishes two qualitatively different kind of contacts,

9

Chapter 2. Fundamentals and Methods

depending on the geometry of the contacting bodies [52]. The first kind is characterizedas complete and occurs whenever the contact area is independent of the force. As anexample, one can consider a flat-ended punch which is pressed into an elastic half-space,as shown in Fig. 2.1 (a). This often leads to difficulties in the analysis of systems whereboth bodies are elastic, because the contact pressure will be singular at the edges of thecontact and the punch cannot be modelled as a semi-infinite body. Therefore, solutionsof incomplete contacts mostly assume a rigid punch. The second kind of contacts ischaracterized as incomplete and occurs when the contact area changes with the loading.These systems are found when two convex bodies come into contact as shown in Fig. 2.1(b). In this case, the two bodies initially only touch in one point. As the applied loadincreases, the contact area increases as well. In contrast to the former case, the contactpressure falls continuously to zero at the edges for incomplete contacts.

p (x)

(a) complete contact

p (x)

(b) incomplete contact

Fig. 2.1: (a) complete contact: contact area is independent from loading. (b) incompletecontact: contact area changes during loading [52]

Moreover, the contact can be assumed as decoupled if the so-called Dundurs’ constantβ is equal to zero [53]. For plane strain β is given as:

β =1

2

1G1

(1 − 2ν1) − 1G2

(1 − 2ν2)1

G1(1 − ν1) + 1

G2(1 − ν2)

, (2.1.1)

where Gi and νi are the shear modulus respectively Poisson’s ratio for the two contactingbodies. The uncoupling condition is satisfied for the following cases [54]:

1. both bodies consist of the same material

2. both bodies are incompressible, ν1 = ν2 = 0.5

3. one body is incompressible and the other body is rigid

4. the contact is frictionless, µ = 0

This condition is also referred to as elastically similar and indicates that the integralsdescribing the normal and tangential loading are uncoupled [52]. As a consequence,variations in the normal force will not induce any tangential displacement and viceversa. Hence, the normal and tangential direction can be analysed separately and theirsuperposition then gives the overall solution. In addition, the initial contact problem

10


of two elastic bodies can be transferred to the indentation of an elastic half-space by arigid punch, where the effective, elastic modules are defined as [55]:

E∗ =

(

1 − ν1

2G1+

1 − ν2

2G2

)−1

and G∗ =

(

2 − ν1

4G1+

2 − ν2

4G2

)−1

. (2.1.2)

Throughout this work, only contacts that are uncoupled and fulfil the half-space as-sumption are considered. Consequently, all systems introduced will be from the lattertype, i.e. indentation of an elastic half-space by a rigid punch. In the following, a fewresults and methods of classical contact mechanics that are important for the furtheranalysis are summarized. Please refer to the well-known books of Popov [55] or Johnson[56] for a detailed representation of the derivations.

2.1.1 Spherical Normal Contact

Consider the classical Hertzian contact [57] of an elastic half-space and a rigid sphere,which is loaded in the z-direction with the normal force FN as shown in Fig. 2.2 (a).The radius of the sphere R is chosen as an effective radius of a contact consisting of twoelastic spheres with particular radii Ri [55]:

R =

(

1

R1+

1

R2

)−1

. (2.1.3)

Instead of solving this initial contact problem, where neither the stress distribution northe area of contact are known initially, one usually considers the deformation of thehalf-space by a given stress distribution. In the classic works of contact mechanics thesedirect solutions are then used as bricks for the solution of the initial contact problem[55].

x

z

2a

d

FN

R

(a) contact problem

p (x, y)

∞

∞

∞

∞y

xz

Ac

(b) direct problem

Fig. 2.2: (a) contact problem: indentation of an elastic half-space by a rigid sphere. (b)direct problem: elastic half-space loaded by a pressure distribution

Sphere and half-space will only touch within a circular area Ac with radius a in thecentre of the contact. This contact area is loaded with the pressure p (x, y) that leads toa vertical displacement of the particles Uz on the surface (z = 0). It is maximal in thecentre (x = 0, y = 0, z = 0), where Uz matches the indentation depth d that is definedas the vertical shift of the sphere. Pressure and normal displacement in the contact

11


area are related through the following integral equation, that is given by the potentialfunctions of Boussinesq and Cerruti [56, 58]:

Uz (x, y) =1

πE∗

∫∫

Ac

p (ξ, η)

sdξdη for x, y ∈ Ac , (2.1.4)

where:

s =√

(ξ − x)2 + (η − y)2 . (2.1.5)

In case of the rigid sphere, the displacement Uz of particles in the contact must matchthe difference of the indentation d and the initial separation h (r), i.e. the profile of thesphere. Using the parabolic approximation1 for R ≫ d [55] this yields:

Uz (r) = d − h (r) = d − r2

2R, (2.1.6)

where r =√

x2 + y2. The following pressure is given by the Hertz theory of the friction-less contact between two bodies of revolution [57]:

p (r) = p0

(

1 −(

r

a

)2)1/2

. (2.1.7)

This pressure fulfils the integral of equation (2.1.4) with Uz as in (2.1.6). Here themaximal pressure p0 and the contact radius a are given as:

p0 =2

πE∗(

d

R

)1/2

, a =√

Rd . (2.1.8)

Finally, the relation between normal force and indentation yields:

FN =4

3E∗R

1/2d3/2 . (2.1.9)

2.1.2 Spherical Tangential Contact

As a consequence of the decoupling assumption, the tangential contact can simply bemodelled as a superposition of the normal contact by a tangential loading FT in thex-direction. Coulomb type of friction with finite constant coefficient µ between thetwo surfaces is assumed. The tangential load is compensated by an initially unknownfrictional shearing traction τ (x, y) on the surface of the half-space. In case that FT isinsufficient to cause gross sliding, i.e. FT < µFN , the contact area will be divided intoa stick region delimited by the stick radius c in the centre and an annular slip regionon the edge, as shown in Fig. 2.3 (a). Inside the stick region, the surface displacement

1The exact profile is given as: h (r) = R −√

R2 − r2.

12


of particles Ux in the direction of FT must be constant, whereas in the slip region, thetraction must match the traction bound:

stick-region: 0 ≤√

x2 + y2 ≤ c ⇒ Ux (x, y) = const. ,

slip-region: c <√

x2 + y2 ≤ a ⇒ τ (x, y) = µp (x, y) .(2.1.10)

The problem corresponds to finding the correct shearing traction distribution that causesthis contact configuration [58]. As for the normal direction, traction and tangentialdisplacement in the contact area are again related through an integral equation, that isgiven by the potential functions of Boussinesq and Cerruti [56, 58]:

Ux (x, y) =1

2πG

∫∫

Ac

τ (x, y)

(

1 − ν

s+ ν

(ξ − x)2

s3

)

dξdη for x, y ∈ Ac . (2.1.11)

Here s is defined as in equation (2.1.5) and G and ν are given as:

G =

(

1

G1+

1

G2

)−1

, ν = G

(

ν1

G1+

ν2

G2

)

. (2.1.12)

Cattaneo [59] and Mindlin [60] independently presented a method of solution for ellipticcontacts for this configuration known as incipient sliding.

a

r

c

Ac

stick

slip

x

y

(a) contact regions

p (r)

τ (r)

p, τ

a rc

(b) traction distribution

Fig. 2.3: (a) contact area with contact radius a and slip-radius c. (b) stress distributionin the contact area

A good description of their procedure is given in book of Popov [55]. The starting pointis the case off full sliding, where the tangential traction must match the traction boundin the entire contact area:

τ (r) = µp0

(

1 −(

r

a

)2)1/2

for 0 ≤ r ≤ a . (2.1.13)

Now, to maintain sticking in case of incipient slip, the traction in the stick region mustbe a superposition of two elliptical distributions of the Hertzian type:

τ (r) = µp0

(

1 −(

r

a

)2)1/2

− τ2

(

1 −(

r

c

)2)1/2

for 0 ≤ r ≤ c . (2.1.14)

13


These tractions cause a parabolic surface displacement in the stick region [55]:

Ux (x, y) = µp0π

32Ga

(

4 (2 − ν) a2 − (4 − 3ν) x2 − (4 − ν) y2)

−τ2π

32Gc

(

4 (2 − ν) c2 − (4 − 3ν) x2 − (4 − ν) y2)

for 0 ≤ r ≤ c , (2.1.15)

that must be constant in accordance to the no slip condition, i.e. Ux = Ustat = const..Finally, this gives the correctional traction-term:

τ2 = µp0c

a. (2.1.16)

The complete traction distribution is as shown in Fig. 2.3 (b). In addition, the stickradius c as well as the constant displacement in the stick region yield:

c = a

(

1 − FT

µFN

)1/3

, (2.1.17)

Ustat = Ux (x, y) = µE∗

G∗ d

(

1 −(

1 − FT

µFN

)2/3

)

for r ≤ c . (2.1.18)

For axially symmetric, three-dimensional contacts the tangential problem can also besolved using the principle of Ciavarella [61, 58] and Jager [62]. According to this, thetangential contact problem can be analysed as one of two normal contacts: the actualnormal contact and an additional normal contact for a reduced load, that provides thecorrection of the tangential traction in the stick area.

2.1.3 Steady State Rolling Contact

Consider a rigid sphere with radius R that rolls on an elastic half-space with shearmodulus G and Poisson’s ratio ν. As depicted in Fig. 2.4 (a), the sphere rotates with anangular velocity ω while it moves to the right with the forward velocity v and is loadedwith FN and FT . Between the surfaces applies dry friction of the Coulomb type withfinite constant coefficient µ. The case v = ωR refers as pure rolling and can only beobtained for infinite friction µ → ∞ or vanishing tangential force2. In case of tractiverolling, i.e. FT 6= 0, occurs a difference between the circumferential velocity and theforward velocity, whose relative ratio is defined as the creep ratio [55]:

ξx =v − ωR

v. (2.1.19)

For ωR > v the friction force on the sphere is pointing forward and the system corre-sponds to a driven wheel. Otherwise, the friction force is pointing backwards and thesystem corresponds to a braking wheel. Reynolds identified creep as the reason for rollingresistance and described that it is caused by both, elastic deformation of the contactingbodies and relative displacement of particles in the interface [63]. The creep tries to drag

2One also distinguishes between free and tractive rolling, i.e. FT = 0 respectively FT 6= 0.

14


the sphere surface particles over the substrate and can be seen as an average amount ofslip. Considering one particle on the surface of the wheel, the contact configuration canbe explained as follows. When entering the contact area at the leading edge, the particleis free of stress. As the sphere rolls forward, it sticks to a particle of the opposing surfaceand is strained by the overall motion difference between the two bodies. This leads to ashear stress, which increases until the traction bound is reached and the particle startsto slip. As for the tangential contact, the contact area is thus divided into a stick and aslip region, the only difference being that the stick area is shifted to the leading edge andthe slip area is shifted to trailing edge. As the state of each surface particle varies overtime, one considers the steady rolling state3 and uses a coordinate system that movesalong with the contact.

FN

FT v

ω

2a

xz

R

2c

(a) tractive rolling of sphere

a

2c

Ac

stick

slip

x

y

(b) contact region

Fig. 2.4: (a) tractive rolling contact of rigid sphere and elastic half-space. The spheremoves with velocity v and rotates with angular velocity ω. (b) contact area with contactradius a and spindle-shaped stick region with half-width c

Hence, in the sticking region, the mass flow of substrate particles that move to the leftwith velocity v must match the mass flow of sphere particles that move to left withvelocity ωR. This gives the relation between creep-ratio and strain as [55]:

ξx =εxx

1 + εxx≈ εxx . (2.1.20)

Again, the normal and tangential problem are decoupled, such that the contact radius 2ais directly given by the Hertzian theory. The stick region is a spindle- or lemon shapedarea of width 2c as shown in Fig. 2.4 (b). One can assume it to be elliptically shaped4

in order to give a closed form solution for the traction and creep [56]. In this case, thesolution procedure resembles to the static problem as in section 2.1.2. Given the relation

3The transient starting process of the rolling is neglected.4This is clearly an error, since the leading edge is located outside the stick region.

15


(2.1.20), the creep ratio5 for the rolling sphere finally results to [64]:

ξx = − 4 − 3ν

4 (1 − ν)

µa

R

(

1 −(

1 − FT

µFN

)1/3

)

, (2.1.21)

and the half-width of the stick region yields:

c = a

(

1 − FT

µFN

)1/3

. (2.1.22)

2.2 The Method of Dimensionality Reduction

Given the approaches introduced in the former sections, incomplete contacts can inprinciple be calculated using a stepwise equilibrium approach on basis of the actualpressure and traction distribution. However, there exist several serious difficulties. Oftenthe exact distribution of pressure remains unknown [55]. Another problem is given bythe different boundary conditions which have to be satisfied for the stick and slip zoneswhen the configuration of these zones is not known in advance [56]. In addition, it oftenremains vague whether a closed form solution of the integrals can be found. Furthermore,numerical simulations, e.g. the boundary element method (BEM), are numerically veryexpensive as the corresponding stiffness matrix is dense.The well-known Method of Dimensionality Reduction (MDR) offers a solution to theseproblems for a variety of applications. The MDR is based on the observation thatclose analogies exist between certain types of three-dimensional contact problems andone-dimensional contacts consisting of independent contact elements. The basic ideawas firstly described in [65] and in the PhD thesis of Geike [66]. A more detaileddescription is given with the book of Popov and Heß[67]. The physical background ofthe MDR lies in the proportionality of the stiffness of a three-dimensional contact to theassociated contact length instead of its contact area [68], what leads to certain mappingrules. Heßderived the exact mapping rules for axial symmetric bodies in various normalcontact configurations including adhesion [69]. If these rules are fulfilled, the macroscopicproperties of the new system coincide exactly with those of the initial system. Thus, theMDR is not an approximation but transfers the initial problem to an equivalent one.According to this, solving contact problems, either analytical or numerical, is trivialized[67]. There is a wide number of applications and further developments of the MDRranging from rough surfaces [70, 71] to viscous media [72], fretting wear [73] and themodelling of stick-slip drives [74]. The MDR is exact for uncoupled (β = 0), rotationalsymmetric, three-dimensional normal and tangential contacts that satisfy the half-spaceassumption [67]. It consists essentially of two basic steps, as illustrated concisely in [75].

5This result is valid only in case that one of the contacting bodies is rigid. If both bodies are fromthe same material the creep ratio is doubled [56].

16


First Step of MDR In this, the three dimensional bodies in contact are replacedby an equivalent elastic or viscoelastic foundation and a rigid indenter. For an elasticcontact, the correct equivalent foundation must consist of linear springs with normalstiffness ∆kz and tangential stiffness ∆kx as shown in Fig. 2.5 (a):

∆kz = E∗∆x and ∆kx = G∗∆x . (2.2.1)

Here ∆x is the distance between adjacent springs, i.e. the grid-size. As the springs areindependent from one another, in the equivalent model remains only one dimension.

E∗, G∗

∞

∞

∞

∞y

x

z

(a) elastic half-space

MDR

∆kz, ∆kx

xz ∆x

(b) one-dimensional system

Fig. 2.5: (a) initial three-dimensional system: elastic half-space. (b) equivalent one-dimensional system: elastic foundation with independent linear springs. First step of MDR

Second Step of MDR In this, the initial three dimensional profile h (r) is transformedinto a one-dimensional profile function g (x) as shown in Fig. 2.6:

g (x) = |x|∫ |x|

0

h′ (r)√x2 − r2

dr . (2.2.2)

MDR

h (r)

g (x)

yx

x

z

z

r

Fig. 2.6: transformation of the three-dimensional profile h (r) into a one-dimensional equiv-alent profile g (x). Second step of MDR

If these rules are fulfilled, the macroscopic properties of the one-dimensional systemexactly match those of the initial three-dimensional system. However, special care mustbe taken not to violate the initial assumptions of the MDR. The following sectionsdescribe the application of the MDR to the tangential contact as well as the steady staterolling contact.

17


2.2.1 Spherical Tangential Contact using MDR

Consider a three-dimensional contact of a rigid sphere with radius R that is pressed intoan elastic half-space by the normal force FN and subsequently loaded in the tangentialdirection with FT . Coulomb friction with constant µ is assumed. In the first stepof the MDR, the elastic half-space is replaced by an equivalent system consisting ofindependent linear springs with normal stiffness ∆kz and tangential stiffness ∆kx asin equation (2.2.1). In the second step of the MDR the approximation of the initialthree-dimensional profile of the sphere as in equation (2.1.6):

h (r) =r2

2R(2.2.3)

is mapped to a one-dimensional profile according to equation (2.2.2):

g (x) = |x|∫ |x|

0

1

R

r√x2 − r2

dr , (2.2.4)

g (x) =x2

R. (2.2.5)

This result can be interpreted as a stretch of the initial profile [76], where the new radiusof the one-dimensional profile R1D yields:

R1D =1

2R . (2.2.6)

y

x

zFN

FT

FT

µE∗, G∗

R

(a) three-dimensional contact

MDR g (x)

xzFN

FT

R1D

(b) one-dimensional system

Fig. 2.7: (a) initial three-dimensional, tangentially loaded, elastic contact. (b) equivalentone-dimensional system consisting of an elastic foundation with stretched radius R1D

Fig. 2.7 gives the initial three-dimensional system (a) and the equivalent one- dimensionalmodel (b). The major advantage of the MDR is that the integral equations with mixedboundary conditions that arise in contact mechanics, are replaced by simple algebraicequations [67]. Instead of a pressure and traction distribution one only needs to know

18


the displacement of the springs in the normal and tangential direction Uz and Ux. Thecorresponding spring forces are then given by the relations:

fz (x) = ∆kzUz (x) and fx (x) = ∆kxUx (x) . (2.2.7)

The missing displacements can be derived by kinematic considerations, as all springs areindependent from one another. In case of the normal direction, the deflection Uz for allsprings in contact is simply determined by the difference between the profile g (x) andthe indentation depth d:

Uz (x) = d − g (x) = d − x2

Rfor − a ≤ x ≤ a . (2.2.8)

Given the condition that Uz must be zero at the contact edges x = ±a and equa-tion (2.2.8) the missing contact radius yields:

Uz (|x| = a) = 0 = d − a2

R⇒ a =

√Rd . (2.2.9)

This corresponds exactly to the well-known result of the Hertzian contact [55], see (2.1.8).The macroscopic normal force FN must match the overall normal force of the springs:

FN =

∫

FdFz . (2.2.10)

With the relation:

dFz = lim∆x→0

∆kz

∆xUzdx = E∗Uzdx (2.2.11)

and equation (2.2.8) this yields:

FN = 2

∫ a

0E∗Uzdx = 2

∫ a

0E∗(

d − x2

R

)

dx , (2.2.12)

FN =4

3E∗R

1/2d3/2 , (2.2.13)

what is again equivalent to the well-known Hertzian contact [55], see (2.1.9). The tan-gential displacement can be derived as follows. For FT insufficient to cause completesliding, i.e. FT < µFN , slipping will only occur at the boundary area of the contact. Inthis region, the forces of the springs equal the traction bound, what in connection withequations (2.2.7) and (2.2.8) gives:

fx (x) = µfz (x) ⇒ Ux (x) = µE∗

G∗

(

d − x2

R

)

. (2.2.14)

The centre region remains in a state of stick and is delimited by the stick radius c. Here,the tangential displacement is constant and matches the displacement between sphereand substrate Ustat. Taken together the tangential contact configuration reads:

stick-region: 0 ≤ |x| ≤ c ⇒ Ux (x) = Ustat ,

slip-region: c < |x| ≤ a ⇒ Ux (x) = µE∗

G∗

(

d − x2

R

)

.(2.2.15)

19


The missing stick radius can be derived from the condition that the displacements mustbe continuous at the edge of the stick region x = ±c. With equation (2.2.15) this yields:

Ustat = Ux (x = c) = µE∗

G∗

(

d − c2

R

)

⇒ c2 = R

(

d − G∗

E∗Ustat

µ

)

. (2.2.16)

The macroscopic force FT must match the overall normal force of the springs:

FT = 2

∫ c

0G∗Ustatdx + 2

∫ a

cµE∗

(

d − x2

R

)

dx . (2.2.17)

Together with (2.2.16) this gives the static displacement as:

Ustat = µE∗

G∗ d

(

1 −(

1 − FT

µFN

)2/3

)

. (2.2.18)

Finally, inserting (2.2.18) into (2.2.16) yields the stick radius as:

c = a

(

1 − FT

FN

)1/3

. (2.2.19)

Again, c and Ustat correspond to the well-known results of the three-dimensional system[55], see (2.1.17) and (2.1.18). This shows that tangentially loaded contacts of a rigidsphere and an elastic half-space can be mapped exactly to an equivalent one-dimensionalsystem. This applies for tangential contacts of arbitrarily shaped bodies as well, as longas the assumptions of axial symmetry and decoupled displacements are not violated [67].

2.2.2 Steady-State Rolling Contact using MDR

In this section it is shown that three dimensional tractive rolling contacts can be mappedto a one-dimensional equivalent system as well. Starting point is the steady state rollingcontact between an elastic half-space and a rigid sphere as introduced in section 2.1.3.The sphere is pressed into the half-space by the normal force FN and is tangentiallyloaded with FT . The sphere moves with velocity v in the x-direction and rotates withan angular velocity ω, as shown in Fig. 2.8 (a).

FN

FTv

ω

xz

(a) rolling contact

MDR

R1D

∆kz, ∆kx

2cζ

2a

stick

slip

(b) equivalent system

Fig. 2.8: (a) steady state rolling contact between a sphere and an elastic half-space. (b)one-dimensional equivalent system with constantly moving coordinate ζ

20


Again, in the first step of MDR, the elastic half-space is replaced by an elastic foundationthat consists of a number of similar independent springs with normal stiffness ∆kz andtangential stiffness ∆kx as in equation (2.2.1). Likewise, the radius of the MDR systemR1D is stretched as R1D = 1/2R according to second step of MDR (2.2.2). With this, theMDR system gives the exact relations between load, contact configuration and rigid bodydisplacement for the normal contact, as shown in section 2.2.1 [67]. Thus, indentationdepth d and contact radius a correspond exactly to the Hertzian contact and one onlyhas to consider the uncoupled tangential problem.As in the static case, the contact region is divided into a stick- and a slip-region. Atthe leading edge at the right side of the contact, the springs are initially completelyrelaxed. As soon as they touch the sphere they start to stick. The tangential force of anarbitrary spring fx then increases with the further rolling due to the elastic displacement.Its normal force fz initially increases as well until the lowest point of the sphere rollspast it. The normal force then decreases until finally fx exceeds the friction boundfT,max = µfz and the spring starts to slip. In consequence, the contact region is dividedinto a stick-region at the leading edge and a slip-region at the trailing edge, as shown inFig. 2.8 (b). For further consideration, the co-moving coordinate ζ is chosen that has itsorigin exactly at the leading edge and moves with the sphere as shown in Fig. 2.8 (b).The tangential displacement Ux (ζ) of an arbitrary spring at position ζ then correspondsto the so far unknown path difference ∆v1t = (ωR − v) t between the sphere and thesubstrate. With ζ = vt it follows:

Ux (ζ) = ∆v1t = ∆v1ζ

vtt =

∆v1

vζ , (2.2.20)

what gives the tangential force for the springs in the stick region 0 ≤ ζ ≤ 2c:

fx (ζ) = ∆kxUx (ζ) = ∆kx∆v1

vζ . (2.2.21)

In the rest of the contact region 2c ≤ ζ ≤ 2a, the springs are slipping:

fx (ζ) = µfz (ζ) = µ∆kzUz (ζ) . (2.2.22)

The missing normal displacement Uz (ζ) is given by the indentation of the rigid sphereinto the elastic foundation of independent springs. With the indentation depth d andthe approximation of the profile of the sphere for R ≫ d as in in equation (2.1.6) follows:

Uz (ζ) = d − (a − ζ)2

2R1=

2aζ − ζ2

R. (2.2.23)

The unknown stick-radius is given by the stick condition at ζ = 2c:

fx (2c) = µfz (2c) ,

∆kx∆v1

v2c = µ∆kz

4ac − 4c2

R,

⇒ c = a − 1

2

G∗

E∗R

µ

∆v1

v. (2.2.24)

21


As the displacement and the contact radii are known, one can determine the macroscopicforce FT via integration. Using equations (2.2.21) - (2.2.24) the tangential force reads:

FT =

∫ 2c

0G∗ ∆v1

vζdζ +

∫ 2a

2cµE∗ 2aζ − ζ2

Rdζ =

4

3

µ

RE∗(

a3 − c3)

. (2.2.25)

Division of equation (2.2.25) by the well-known Hertzian relation of equation (2.1.9)initially yields:

FT

FN= µ

(

1 −(

c

a

)3)

. (2.2.26)

Rearranging this equation, one obtains the same expression for the stick-radius as forthe initial three-dimensional case as in equation (2.1.22) or shown in [55]:

c

a=

(

1 − FT

µFN

)1/3

. (2.2.27)

In addition rearranging (2.2.24) and inserting (2.2.27) gives the creep ratio ξ1 of theone-dimensional MDR model:

ξ1 = −∆v1

v= −1

2

E∗

G∗µa

R

(

1 −(

1 − FT

µFN

)1/3

)

. (2.2.28)

Given the relations for the effective modules [55]:

E∗ =2G

1 − νand G∗ =

4G

2 − ν, (2.2.29)

one can compare the one-dimensional creep with the original creep ξx as stated in equa-tion (2.1.21):

ξx =v − ωR

v= − 4 − 3ν

4 (1 − ν)

µa

R

(

1 −(

1 − FT

µFN

)1/3

)

. (2.2.30)

This yields the following relation between the creep ratios:

ξ1 =4 (2 − ν)

(4 − 3ν)ξx . (2.2.31)

Equation (2.2.31) shows that the result of the one-dimensional system must be scaled inorder to determine the correct creep of the three-dimensional system. The deviation isdue to the asymmetry in the tangential stress distribution of the rolling contact [55, 56]that contradicts the MDR assumption of rotational symmetry of the contact configura-tion. In summary it shows that the MDR enables the mapping of the three-dimensionaltractive rolling contact to a one-dimensional equivalent system. Thus the MDR allowsto analyse more complex problems, such as the oscillating rolling contact as introducedin section 3.1. However, it must be ensured that the influence of potential contradictionsto the MDR assumptions are cautiously taken into account.

22


2.3 Modelling of Frictional Systems

In this section some important aspects considering the modelling of frictional systemsare briefly described. The principles introduced will be used in the following chapters.

2.3.1 Separation of Scales

As mentioned in the introduction of this chapter, one important characteristic of frictionis the interaction between various factors as for instance the local deformation in thecontact and the macroscopic motion of the system. However, under certain assumptionsit is possible to separate certain time and space scales that contribute to the system.

Quasi-Statics The main requirement is quasi-statics of the contact problem [55]. Acontact can be assumed to be quasi-static as long as its characteristic time of loading Tc

is much larger than the time an elastic wave in the continuum needs to travel a distanceof the order of the contact length Dc [67]:

Tc ≫ Dc

cs, (2.3.1)

where cs denotes the speed of sound of the material. This is fulfilled in most tribologicsystems, as the speed of sound of the materials is much larger than the relative movementof components, which determines the contact time Tc. In consequence, the contactsinstantaneously reach the equilibrium state. Thus, even in case of an unsteady loadingone can apply static equilibrium conditions in every time step to calculate the forces anddeformations in the contact.

Elastic Energy The linear elastic deformations in the contact decrease with 1/r, withr being the distance to the contact area. In addition, the stresses and strains decreasein proportion to 1/r2. Therefore, the elastic energy of the deformation is concentratedin a volume with linear dimensions of the order of the contact width Dc [65]. The elasticenergy of the contacts can thus be characterized as a local quantity that only dependson the conditions at the vicinity of the contact. In consequence it is independent fromthe size of the macroscopic body as well as other contacts of the system, as long as thedistance between adjacent contacts is larger than their contact length [55].

Kinetic Energy In contrast, under most conditions, the kinetic energy of systemsinvolving frictional contacts is a non-local quantity that is independent from the contactconfiguration. The kinetic energy of the system as a whole corresponds to the kineticenergy of the bulk body. Thus, the inertial properties of the system can be sufficientlydescribed by the rigid body dynamics of the bulk body[65].

It should be emphasized that the elastic and kinetic energy of the system are not trulyindependent as both are connected through the forces and moments that act in the

23


contact and on the rigid body, i.e. Newton’s third law. But both are decoupled throughthe separation of space scales. In case of dynamic problems this gives the possibility todetermine the rigid body motion using the equations of motion and then solve for thecontact configuration in every time step. In turn the resulting forces and moments serveas input parameters for determination of the next step of the rigid body motion.

2.3.2 Modelling of Coulomb Friction in Dynamic Systems

Once the static maximum holding force FT,max = µsFN is exceeded and sliding hasstarted, the friction force drops to its lower kinetic value FR = µkFN , where µk < µs.In addition, it was shown in numerous experiments that the kinetic friction is a functionof the relative velocity v with a negative slope [77, 78, 9, 79]. This phenomenon is alsoreferred to as rate weakening [38]. Both effects lead to a non-linear behaviour of mechan-ical systems with dry friction. A stick-slip motion may occur where the sliding surfacesalternately switch between sticking and slipping [3]. In the modelling, the distinctioninitially leads to a set of differential equations with discontinuous right hand side. Onewidespread solution is to apply a smoothing method for the frictional coefficient [9, 80]what yields a system of ordinary differential equations. A popular approach is to applya velocity dependent coefficient as shown in Fig. 2.9 [81]:

µk (v) = µs2

π

arctan (γv)

1 + |v|δv

, (2.3.2)

where v denotes the relative velocity between the surfaces. In equation (2.3.2), the limit-ing velocity δv defines the velocity range in which static friction applies and γ determinesthe steepness around v = 0. However, this type of model has several drawbacks. Firstly,the unknown and in principle arbitrary parameters δv and γ need to be determinedand there is no standard rule for their estimation. Secondly, this type of model cannotaccurately represent sticking, since it allows a nearly stationary system, i.e. v → 0,to accelerate even if the net tangential forces are less than the maximum holding force[38, 82]. This was also examined in detail by Elmer who states that any kinetic frictionlaw as in (2.3.2) becomes invalid if:

v <Dc

Tc, (2.3.3)

where Dc denotes the microscopic length scale and Tc the macroscopic time scale, i.e. thecontact time [83]. Increasing the steepness parameter γ will improve the approximationaround v = 0. However, this will also lead to large computation times as the differentialequations become stiff, particularly in case of large frictional resistance. Another popularmodel is the so called force-balance method developed by Karnopp [84]. Here, for v > δvthe friction force is a function of the sliding velocity. For v < δv the friction force isdetermined as such that it balances all other forces in the tangential direction in case ofsticking, i.e. leads to zero acceleration, or matches the friction bound in case of slipping.An important extension of the force balance method is the switch model developed by

24


Leine et al. [81]. On basis of the velocity and the external forces it is determinedwhether the system slips or sticks and the corresponding time derivatives are chosen,i.e. the differential equation is changed. One drawback of the models introduced istheir inability to describe so-called pre-sliding, an effect that is caused by the elasticdeformation of the contact whenever a sticking system is loaded tangentially. This iseffect is captured by the elastoplastic friction model [85] which is an extension of thecommonly used Dahl-model [86]. However, the model consists of a number of empiricalparameters which are difficult to determine. Some can be measured while others have tobe chosen arbitrarily, so that they are more likely fitting parameters by nature [74]. Inaddition, Oden and Martins conclude that the exact slope of the friction function is notdefined uniquely by the nature of the surfaces in contact, but is rather a consequence ofall the dynamic variables involved [9]. Thus it is not only affected by the slip velocitybut also by inertia and stiffness properties of the frictional system. Finally, a clearunderstanding of frictional sliding phenomena at small speeds has not been yet achievedand none of the models proposed has gained general acceptance [10].

µk (v)

v

µs

−µs

Fig. 2.9: smooth model function of ve-locity dependent coefficient of friction

X

Z

kxkz

Ux

Uz

µ

Fig. 2.10: rigid slider with elastic con-tact modelled as a linear spring

2.3.3 Algorithm for the Modelling of Contacts

The objective of this study is in particular to identify the influence of microscopic effects,e.g. the spatial variations of pressure and traction and the distribution of stick and slipin the contact interface, on the macroscopic behaviour of the system. Parameters thatcan be rather characterized as empirical, e.g. γ or δv, complicate the clear interpretationof the results. Thus, it is convenient to use a relatively simple friction model for thecontact itself in order to separate the influences of system dynamics. This corresponds tothe approach of Martins et al. [10]. Their model neglects the distinction between staticand kinetic case and the dependency on the relative velocity. However, they were able tosimulate apparently smooth sliding motions with apparent coefficients of kinetic frictionsmaller than the static one and also stick-slip oscillations. Hence, their relatively simplecontact model and the interplay with the system dynamics reflects the experimentallyobserved dependencies on the velocity, although these dependencies are not consideredin the initial contact model, i.e. µ is not a function of time or velocity.

25


Likewise in our model, the elastic deformations of the contact are taken into accountwithout distinction between static and kinetic coefficient of friction. The model that isused in this work and its corresponding equations of motions are described in detail insection 4.1. In order to explain the basic principle, the rigid slider with two degrees offreedom X and Z is introduced, which is depicted in Fig. 2.10. On basis of the separationof scales approach as described in section 2.3.1 the contact is modelled as a linear springwith normal stiffness kz and tangential stiffness kx and corresponding spring deflectionsUz and Ux. Thus, the spring forces are given as:

Fz = kzUz and Fx = kxUx . (2.3.4)

Coulomb’s law of dry friction applies in the contact with a microscopic coefficient offriction µ. Therefore the contact sticks whenever:

|Fx (t) | ≤ µFz ⇔ kx|Ux (t) | ≤ µkzUz (t) . (2.3.5)

The deflections are dependent variables that are directly determined by the motion ofthe system, i.e. they depend on the translation X and Z. The time derivative of thenormal spring deflection equals the normal velocity:

Uz (t) = Z (t) . (2.3.6)

In accordance to the switch model [81] the friction force FR and the tangential springdeflection are given by a case distinction. For this purpose a test deflection is calculatedassuming sticking of the contact:

Ux (t) = Ux (t − ∆t) + ∆tX (t − ∆t) . (2.3.7)

Here ∆t denotes the time step. Thus, in case of sticking the rate of tangential deflectionUx corresponds to the velocity of the mass X. Finally, the case distinction reads:

kx|Ux (t) | ≤ µkzUz (t) ⇒ sticking ⇒ FR (t) = −kxUx (t) ,

⇒ Ux (t) = Ux (t) ,

kx|Ux (t) | > µkzUz (t) ⇒ slipping ⇒ FR (t) = −µ sign(

Ux (t))

kzUz (t) ,

⇒ Ux (t) = µ sign(

Ux (t))

kzkx

Uz (t) .

(2.3.8)

This algorithm ensures continuity of the so called state vector of the system withoutnumerical instabilities and enables pre-sliding, as the rigid body can move tangentiallyeven for sticking contacts. In a slightly modified form, this friction model is used forthe analysis of systems with sliding friction in section 4. The modifications enable toconsider the cases of completely released contacts, a spatial variation of stick and slipand negative spring deflections in the tangential direction. A detailed description of themodel used in the analysis is given in section 4.1. In addition, the contact model isdescribed in detail in appendix B.4.

26

Chapter 3

Shakedown and Ratcheting in

Incomplete Contacts

Oscillating micro-slip can cause failure of frictional contacts in nominally static systems,i.e. systems that are intended to stay in a stable equilibrium without macroscopic slip.One has to distinguish three cases. Firstly, the system can shakedown what means thatthe slip ceases after the first few periods. Secondly, ratcheting can occur, where themicro-slip accumulates and leads to a continuing body motion. Thirdly, reversing slipmight occur, what describes the situation in which no net tangential motion occurs. Inthe following chapter a model for a nominally static system is introduced, which consistsof an incomplete contact, meaning that the contact area changes during oscillation. Theexact shakedown limits are derived. These are the prerequisites that must be fulfilled inorder for the system to shakedown. In addition, the supercritical case of ratcheting isexamined. Finally, some variations of the initial system are considered.

3.1 Oscillating Rolling Contact

In order to give a generic model for a force locked connection with an incomplete con-tact, the so called oscillating rolling contact is introduced. Starting point is the tan-gentially loaded contact between a rigid sphere and an elastic substrate as described insection 2.1.1 and section 2.1.2. A good description for this system can also be found forexample in [55] or [56]. As introduced before, the elastic modules E∗ and G∗ of the sub-strate, i.e. of the half-space, as well as the radius of the sphere R are effective quantitiesof a contact of two elastic spheres. The system is assumed to be uncoupled, meaningthat variations in the normal force will not induce any tangential displacement and viceversa, i.e. Dundurs’ constant β = 0 applies. Dry friction of the Coulomb type withconstant coefficient µ between sphere and substrate is assumed. The sphere is pressedon the substrate by the normal force FN and is additionally loaded with a tangentialforce FT . The normal force leads to the indentation depth d, which is defined as the

27

Chapter 3. Shakedown and Ratcheting in Incomplete Contacts

vertical displacement of the rigid sphere:

d =

(

3FN

4E∗R1

2

)2/3

. (3.1.1)

The area of contact is delimited by the contact radius a =√

Rd . A tangential loadingFT , insufficient to cause complete sliding, will lead to a slight rigid body displacementof the substrate in the direction of the force:

Ustat = µE∗

G∗ d

(

1 −(

1 − FT

µFN

)2/3

)

. (3.1.2)

This static displacement refers to the relative shift of points that are sufficiently far awayfrom the contact area, i.e. it equals the constant surface displacement as in (2.1.18). Asshown in section 2.1.2 slipping will initially only occur at the boundary region of thecontact area, whereas the centre region remains in a state of stick and is delimited bythe stick-radius c as in (2.1.17):

c = a

(

1 − FT

µFN

)1/3

. (3.1.3)

In the next step, this static tangential contact is superposed by a slight oscillatory rollingof the sphere with amplitude W , being the lateral movement of the centre of the sphere,as depicted in Fig. 3.1 (a). The system is assumed to be quasi-static, meaning that aconstant µ is assumed and inertia effects are neglected, as described section 2.3.1. Thisis valid as long as the excitation W is slower than the propagation speed of elastic wavescs within the bodies, i.e. W ≪ cs.

FNFN

FT

FTµ = 0

W

W

xz

U substrate

(a) oscillating rolling contact

W

tT 2T

(b) time response of the rolling

Fig. 3.1: (a) oscillating rolling contact between rigid sphere and elastic substrate. Move-ment of the centre of the sphere is denoted by W and displacement of the substrate isdenoted by U . (b) time response of the rolling W with oscillation period T

The overall macroscopic normal and tangential forces will both be kept constant in mag-nitude and the rolling will not induce any additional forces or moments. However, therolling varies the contact area and leads to a periodic change of the pressure distribution.

28


As depicted in Fig. 3.2 the actual trailing edge is released and the corresponding fric-tion bound τmax = µp is reduced. According to this, the problem setting is equivalentto a frictional contact, which is exposed to an oscillatory back and forth movement ofthe normal force. This corresponds to a rocking motion of the contacting bodies. Thesystem is thus the three-dimensional Hertzian counterpart of the rocking and walkingpunch [27, 28, 29] and represents a general model of force-locked connections where thecontact area changes under the influence of vibrations.

FN =const. FN =const.

FT =const. FT =const.

τmaxτmax releaserelease

Fig. 3.2: oscillating rolling of the sphere releases the actual trailing edge and alternatelyreduces the friction bound τmax

3.1.1 Influence of the Oscillating Rolling

In order to determine the influence of the oscillating rolling, the displacement of thesubstrate U is examined. The essential factors of influence are the tangential force FT

and the rolling amplitude W . All three variables are normalized as:

u =U

Umax, fT =

FT

FT,max, w =

W

a. (3.1.4)

Here FT,max equals the maximal holding force of equation (1.0.1) and Umax describesthe static displacement in equation (3.1.2) for FT = FT,max:

Umax = µE∗

G∗ d . (3.1.5)

Only tangential forces below the maximum holding force and oscillation amplitudessmaller than the contact radius are considered:

fT ≤ 1 and w ≤ 1 . (3.1.6)

Consequently, without oscillatory rolling, no complete sliding will occur and the centreof the sphere will not be moved beyond the initial area of contact at any time. Thelatter means that there will always remain an area in the centre of the contact that isnever released. Thus, taken by itself, neither of the two factors leads to a failure of thecontact.Figure 3.3 and Fig. 3.4 give the displacement u over rolling periods for fT = 0.09 andfT = 0.72 and different amplitudes w. These time responses are the product of theCONTACT simulation that is described in appendix B.2. They agree well with theexperimental ones, that are shown in section 5.1.4. Both, simulation and experiment

29


show that the oscillatory rolling increases the displacement of the substrate with respectto the static value ustat. For fT and w being sufficiently small, the displacement stops andthe system reaches a new equilibrium as shown in Fig. 3.3. This effect is characterizedas a frictional shakedown and is explained in more detail in section 3.2. Otherwise, if fT

and w exceed certain limits, the displacement continues and so called ratcheting occurs.This case is shown in Fig. 3.4 and also in 3.3 for w = 0.74 and is discussed in section 3.3.Comparison of both cases shows that reaching the shakedown equilibrium requires morecycles than reaching a steady state ratcheting state.

0 2 4 6 8 10 12

0.05

0.1

0.15

0.2

ustat

usd

periods n

dis

pla

cem

ent

u

w = 0.24w = 0.51w = 0.74

Fig. 3.3: displacement of the substrateu for fT = 0.09 and different oscillationamplitudes w. Shakedown: displacementstops and system reaches a new equilib-rium

0 2 4 6 8 10 12

2

4

6

8

ustat

periods n

dis

pla

cem

ent

u

w = 0.24w = 0.51w = 0.74

Fig. 3.4: displacement of the substrateu for fT = 0.72 and different oscillationamplitudes w. Ratcheting: displacementcontinues with the rolling

3.1.2 MDR Model of the Oscillating Rolling Contact

In order to describe the introduced system, one can theoretically use the classical ap-proach of Cattaneo [59] and Mindlin [60], as in described in section 2.1.2. For thispurpose both the contact configuration, i.e. the distribution of stick and slip, as well asthe distribution of stress must be known. Then the potential functions of Boussinesqand Cerruti [56] determine the tangential displacement of the substrate. It is at leastdoubtful whether this can be achieved for the given system, with its changing contactarea and stress field.The results of section 2.2.2 prove the usefulness of the MDR in the modelling of rollingcontact problems. Therefore, the MDR is the method of choice for the description of theoscillating rolling contact. Again the profile of the sphere is transformed in accordanceto the second step of the MDR, see equation (2.2.2):

R1D =1

2R . (3.1.7)

And the elastic substrate is replaced by an elastic foundation of independent, linearsprings with normal and tangential stiffness ∆kz and ∆kx as stated in the first step of

30


the MDR, see equation (2.2.1):

∆kz = E∗∆x and ∆kx = G∗∆x . (3.1.8)

Both the initial model and the MDR model are shown in Fig. 3.5. One major advantageof the MDR is that due to the independence of the springs, the corresponding deflectionsof the springs can be directly determined by kinematic considerations. Consequently,the instantaneous displacement in the z-direction at the reversal points of the rollingyields:

Uz (x) = d − (x ∓ W )2

2R1D= d − (x ∓ W )2

R, (3.1.9)

where x gives the position of one particular spring. Using this relation one can derivethe analytical description of the system. In addition, one can perform a quasi-staticincremental simulation as described in more detail in appendix B.1.

FN

FT

FT

R

(a) three-dimensional contact

MDR

FN

FT

FT

µ = 0 U

x

zR1D

∆kz, ∆kx

(b) equivalent MDR model

Fig. 3.5: (a) three-dimensional tangential contact. (b) equivalent one-dimensional MDRmodel with mapped radius R1D

3.2 Shakedown of the Oscillating Rolling Contact

The characteristic response in case of shakedown, which is shown in Fig. 3.3, can beexplained as follows. At first, the system is in static equilibrium due to the constantmacroscopic forces. As fT < 1 the displacement of the substrate corresponds to the staticvalue u = ustat of equation (3.1.2). The back and forth rolling of the sphere then altersthe pressure distribution as depicted in Fig. 3.6. When the sphere starts to roll forward,the pressure at the trailing edge drops. As this reduces the friction bound τmax = µpslipping occurs and the traction decreases at the trailing edge. At the same time, atthe leading edge, the pressure increases but the traction initially remains constant. Thisresults in an imbalance of force in the tangential direction. The same effect occurs, whenthe sphere rolls backwards. Thus, at first, this reduces the stick area and increases therigid body displacement between sphere and substrate u within every back and forthmovement of the sphere, as shown in Fig. 3.3. However, for sufficiently small fT and w asaturation level is reached after a certain number of periods and the displacement stops.The residual force within the contact is sufficiently strong to prevent any further slip

31


and the system reaches a new equilibrium, i.e. the shakedown state. The displacementthen corresponds to the constant, time independent shakedown displacement u = usd

[25].

p(x

,0)

p(x

,0)

W

W −W

−W

2a2axx

Fig. 3.6: alternating contact area and periodic displacement of the pressure distributionp (x, y = 0) as a consequence of the rolling motion with amplitude W

slip

stick +W −WShakedown

n = 0 n = 0.25 n = 2.75 n = 12

Fig. 3.7: distribution of stick (black) and slip (grey) area in case of shakedown after variousnumbers of rolling periods n for w = 0.24 and fT = 0.56 (CONTACT Simulation). Slippingoccurs on the actual trailing edge while the actual leading edge sticks. After several periodsthe system reaches a new equilibrium

Figure 3.7 depicts the time evolution of the contact area. In the beginning the contactarea is divided into the stick and slip area1. Due to the rolling, slipping2 occurs on theactual trailing edge while the actual leading edge sticks. In the centre of the contact arearemains a region that permanently sticks. After several periods a shakedown occurs.Even if the rolling is continued, the whole contact area remains in a state of stick,respectively in a state of no-slip in case of the regions that are released periodically.Another way to enlighten the transient process towards the shakedown state is to examinethe frictional dissipation in one time step Wfric. A frictional shakedown is then and onlythen reached if Wfric vanishes. Hence, so called trailing edge slip may not occur, as thiswould refer to the case of reversing slip. The frictional dissipation is defined as theintegral of the traction τ (x, y) times the local slip displacement s (x, y) in one time step

1This means that the friction bound τmax = µp is reached.2This refers to a relative motion between particles.

32


over the slip area Aslip [87]:

Wfric =

∫

Aslip

τ (x, y) s (x, y)dA . (3.2.1)

The CONTACT model with world fixed (space-fixed) approach and 55 x 45 quadraticdiscretization elements with a grid-size of ∆x, ∆y = 0.22 mm and the parameters of theexperimental setting as described in 5.1.1 is used to compute Wfric. The dissipation ofone cycle of an oscillating tangential loading with amplitude FT [56]:

W0 =18µ2F 2

N

G∗5a

(

1 −(

1 − FT

µFN

)5/3

− 5

6

FT

µFN

(

1 −(

FT

µFN

)2/3

))

. (3.2.2)

is used for normalization. The results consistently show, that the frictional dissipationdecreases rapidly after the first few periods, as depicted in Fig. 3.8.

10 20 3010−12

10−9

10−6

10−3

periods n

dis

sipat

ion

Wf

ric

/W0 ε = 10−7

ε = 10−9

ε = 10−12

Fig. 3.8: normalized dissipation pertime step Wfric/W0 for different relativeaccuracies ε in case of shakedown

5 10 15 20−2

0

2

4

·10−7

periods n

rigi

dsh

ift

∆U

s/U

ma

x ε = 10−7

ε = 10−9

ε = 10−12

Fig. 3.9: normalized rigid body shift pertime step Us/Umax for different relativeaccuracies ε in case of shakedown

Afterwards for n > 5, dissipation and thus trailing edge slip only occurs exactly in thereversal points of the rolling motion. The dissipation then decreases constantly with eachhalf period and finally reaches a level, that depends on the requested relative accuracyof the output variables ε. The normalized dissipation is in the range of 10−7 − 10−13 atthe end of the process3. The corresponding rigid body shift per time step ∆Us is givenin Fig. 3.9 and shows that the dissipation in the steady state is caused by a numericaleffect. There occurs a characteristic back and forth motion in the region of constantdissipation, which lacks any physical background. A constant tangential force, as it isthe case here, cannot produce a reciprocating motion in the tangential direction. Besides,for elastically similar materials (β ≈ 0), slip always occurs in the same direction, that

3For the parameters used in the simulation this corresponds to 2 · 10−14 J, i.e. the rest mass-energyof elementary particles.

33


is given by the tangential load [29]. Thus reversing slip cannot occur as fT = const..Additionally, the amplitude of the shift is quite small again4. One can conclude thatboth, the characteristic motion and the according dissipation, are most likely caused bythe iteration process. It is based on a Newton-Raphson procedure [88], which does notimprove the solution any more, when the requested relative accuracy is reached. Thiscorresponds exactly to the system behaviour in the steady state as shown in Fig. 3.8 andFig. 3.9 for n > 10. Taking into account the arguments above, the residual dissipationand rigid body shift can be neglected. Consequently, neither slipping nor dissipation willoccur when the new equilibrium has been reached. One can conclude that the systemresponse is a true frictional shakedown. Reversing slip will not occur as the direction ofthe tangential force does not change during the oscillation [29]. Figure 3.9 shows thatthe new equilibrium is reached after n ≈ 10 cycles. After this, the only displacement isthe non-physical back and forth motion.

3.2.1 Contact Configuration after Shakedown

In order to analyse the contact configuration in the new equilibrium, the CONTACTmodel with world fixed approach, see appendix B.2, and 108 x 89 quadratic discretizationelements with a side length of ∆x, ∆y = 0.11 mm and the parameters of the experimen-tal setting as described in section 5.1.1 in Tab. 5.1.1 is used. Figure 3.10 depicts thenormalized traction τ (x, y) /µp0 after shakedown, where p0 denotes the static maximumpressure at (x, y) = (0, 0) [55]:

p0 =2

πE∗(

d

R

)1/2

. (3.2.3)

−1

0

1

−1

0

10

0.5

1

x/ay/a

τ(x

,y)/µ

p 0

Fig. 3.10: traction τ (x, y) /µp0 aftershakedown for w = 0.24 and fT = 0.56(CONTACT model)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

stickregion

slipregion

zerotract.

coordinate x/a

trac

tion

τ(x

,0)/µ

p 0

Fig. 3.11: traction τ (x, y = 0) /µp0 af-ter shakedown at the centre line alignedparallel to the rolling for w = 0.24 andfT = 0.56 (CONTACT model)

4This corresponds to 5 nm for the experimental parameters.

34


It shows that in the new equilibrium, there occur three characteristic contact regions.These are illustrated in Fig. 3.11 where the normalized traction is shown along the centreline aligned parallel to the rolling, i.e. τ (x, y = 0). In the centre occurs the stick region,where the displacement of particles ux must match the shakedown displacement usd inorder to meet the sticking condition. Adjacent thereto occurs the slip region, wherethe tangential stress equals the traction bound that appears at the reversal point of therolling. Finally, at the outside of the contact, occurs the zero traction region. Here, thetraction vanishes as the sphere is periodically released and no trailing edge slip occursin shakedown state. The total contact after shakedown is configured as follows:

stick-region: Ux = const. ,slip-region: τ = µp ,

zero-traction: τ = 0 .(3.2.4)

The according normalized tangential displacement of particles in the contact regionux (x, y) = Ux (x, y) /Umax is depicted in Fig. 3.12. It shows that the stick region inthe centre is slightly spindle shaped, see also Fig. 3.15. Figure 3.13 gives the centreline-displacement of particles ux (x, y = 0). In the stick region the displacement ux corre-sponds to the constant shakedown displacement usd.

−1

0

1

−1

0

10

0.5

x/ay/a

ux(x

,y)

Fig. 3.12: tangential displacementux (x, y) after shakedown for w = 0.24and fT = 0.56 (CONTACT model)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

stickregion

slipregion

zero

tract.

ux = usd

coordinate x/a

dis

pla

cem

ent

ux(x

,0)

Fig. 3.13: tangential displacementux (x, y = 0) after shakedown at the cen-tre line aligned parallel to the rolling forw = 0.24 and fT = 0.56 (CONTACTmodel)

The displacement of particles ux (x, y) for a given traction τ (x, y) and contact area isdetermined by the potential functions of Boussinesq and Cerruti [56]. However, even ifthe exact distribution of τ (x, y) is identified, it remains vague whether a closed solutionof the integral of this classical approach exists. Instead, the MDR model as introducedin section 3.1.2 is used, where the displacements can easily be identified from kinematicrelations. Figure 3.14 shows ux (x) for the MDR model. The blue line symbolizes thestatic displacement before shakedown and the red line the displacement after shakedown.

35


As for the initial three dimensional case, there remain three contact regions. The stickregion in the centre is delimited by the stick radius after shakedown csd. Here ux matchesthe shakedown displacement usd, which is increased in comparison to the static valueustat. Adjacent to the stick region occurs the slip region which is delimited by the slipradius b. In this region, the tangential spring forces match the lowest friction boundof the process, which occurs exactly at the reversal points of the rolling. Finally, atthe outside of the contact area occurs a region in which the displacement and hencethe tangential force is zero due to the alternating release of the springs. In order todetermine the contact configuration, it is briefly returned to the dimensionful notation.Here, the normal and tangential spring forces fz and fx are given as:

fz (x) = ∆kzUz (x) and fx (x) = ∆kxUx (x) . (3.2.5)

The minimal normal displacement is reached in the reversal point of the rolling:

Uz (x) = d − (x ∓ W )2

Rfor − a ± W ≤ x < a ± W . (3.2.6)

Thus, together with the friction bound fx (x) = µfz (x) and equation (3.2.5) the tan-gential displacement in the slip region yields:

Ux (x) = µE∗

G∗ Uz (x) = µE∗

G∗

(

d − (x + W )2

R

)

for csd ≤ |x| < b . (3.2.7)

At position x = csd this must exactly match the shakedown displacement what deter-mines the missing stick radius:

Usd = Ux (x = csd) , (3.2.8)

Usd = µE∗

G∗

(

d − (csd + W )2

R

)

, (3.2.9)

⇔ csd =

√

(

d − G∗

µE∗ Usd

)

R − W . (3.2.10)

The slip radius directly depends on the rolling amplitude and is given as:

b = a − W . (3.2.11)

In summary the contact configuration after shakedown for the MDR system reads:

stick-region: 0 ≤ |x| ≤ csd ⇒ Ux (x) = Usd ,

slip-region: csd < |x| ≤ b ⇒ Ux (x) = µE∗

G∗

(

d − (x+W )2

R

)

,

zero-force: b < |x| ≤ a ⇒ Ux (x) = 0 .

(3.2.12)

36


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

csd/a

b/a

ux = ustat

ux = usd

coordinate x/a

dis

pla

cem

ent

ux

(x) ux−stat

ux−sd

Fig. 3.14: displacement ux (x) beforeand after shakedown for w = 0.24 andfT = 0.56 (MDR model)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

coordinate x/a

coor

din

ate

y/a

0

0.2

0.4

ux

Fig. 3.15: displacement ux (x, y) aftershakedown for w = 0.24 and fT = 0.56(CONTACT model)

With the complete contact configuration being known, it is possible to determine theequilibrium condition. After shakedown, the external tangential force FT must matchthe overall tangential force of the springs:

FT =

∫

FdFx =

∫

Lc

G∗Ux (x)dx , (3.2.13)

where Lc denotes the contact length. With the symmetric contact configuration thisyields the following integral:

FT = 2

∫ csd

0G∗Usddx + 2

∫ b

csd

µE∗(

d − (x + W )2

R

)

dx . (3.2.14)

Finally, inserting equations (3.2.10) and (3.2.11) into (3.2.14) gives the equilibrium con-dition in normalized form:

fT = 1 − 3

2wusd − (1 − usd)

3/2 . (3.2.15)

Equation (3.2.15) specifies the relation between tangential load fT , oscillation amplitudew and shakedown displacement usd. Thus, on the one hand it enables to determine usd

for a given combination of fT and w using numerical solving procedures. On the otherhand, it is possible to derive the shakedown limits, i.e. the maximal values for fT andw for which a safe shakedown occurs. Both is shown in section 3.2.2.However, the equilibrium condition (3.2.15) is the direct result of the MDR, whichis based on the assumption of rotational symmetry of the contact region [67]. Fig-ure 3.15 gives the tangential displacement in the contact region computed with thethree-dimensional simulation. The contour lines show that the distribution of tangentialdisplacement is slightly elliptic or spindle shaped. This can be seen especially at thestick region of constant displacement in the centre. Consequently, a small deviation in

37


the relation between loading and oscillation amplitude of equation (3.2.15) is expected.Considering this, the mapping parameter κ is introduced, such that:

fT = 1 − κwusd − (1 − usd)3/2 . (3.2.16)

3.2.2 Shakedown Limits

Expression (3.2.16) enables the prediction of the shakedown displacement usd for a givencombination of fT and w below the shakedown limits. The unknown mapping parameterκ is gained via comparison with the results of the experiments, where the goodness of thefitting procedure is given in appendix A.1.3 in Tab. A.1.1. The fitting indicates κ = 1.Finally, this yields the following expression for the relation of shakedown displacement,tangential force and oscillation amplitude:

fT = 1 − wusd − (1 − usd)3/2 . (3.2.17)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

usd − ustat

ustat

ratcheting

tangential force fT

shak

edow

ndis

pla

cem

ent

usd

w = 0w = 0.24w = 0.51w = 0.74ulim

Fig. 3.16: shakedown displacement usd as a function of the tangential force fT for differ-ent amplitudes w determined with equation (3.2.17). The oscillating rolling increases thedisplacement with respect to the static value ustat

Figure 3.16 illustrates usd as a function of fT for different w. The solid lines show theanalytical results of (3.2.17), whereas the experimental results are indicated by the error-bars and marks. The asterisks depict the results for the three-dimensional CONTACTsimulation, where values for tangential forces close to the shakedown limit are not given,because the iterative solution procedure of CONTACT lacks robustness in this case [88].As mentioned before, it shows that usd is increased in comparison with the static value

38


ustat. It is also known that in the case that the oscillating rolling stops to soon, the finaldisplacement might differ from this theoretical shakedown value [25]. In the experiments,the shakedown state is already reached after n ≈ 10 rolling periods. This corresponds tothe simulation results shown in the beginning of section 3.2. The dotted line in Fig. 3.16indicates the maximal displacement for different amplitudes that is achieved, before theoscillation leads to a complete failure of the contact. In this case, the stick radius csd

is zero, which in combination with equations (3.2.10) and (3.2.17) gives the maximaltangential force:

fT,lim = 1 − wlim , (3.2.18)

and the maximal displacement:

ulim = 2fT,lim − f2T,lim . (3.2.19)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

tangential force fT,lim

amplitu

de

wli

m

Fig. 3.17: maximal amplitude wlim asa function of the tangential force fT,lim

determined with equation (3.2.18)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


dis

pla

cem

ent

uli

m

Fig. 3.18: maximal displacement ulim

as a function of the tangential forcefT,lim determined with equation (3.2.19)

Figure 3.17 and Fig. 3.18 give the analytical (solid lines) and experimental (error-barsand marks) results for the shakedown limits. For medium tangential forces, both wlim

and ulim show strong agreement with the theory. Fig. 3.17 gives wlim = 1 − fT,lim sincein the experiments, the maximal amplitude is identified by increasing it stepwise whilefT is kept constant. The exact experimental procedure for the determination of theshakedown limits is described in section 5.1.4.These maximal values correspond to the analytical shakedown limits for the oscillatingrolling contact. Thus, for a given oscillation amplitude w, the maximum tangential forcefT,lim to achieve a safe shakedown can be identified and vice versa. Additionally, sincefT,lim ≤ 1, it turns out that in case of the oscillatory elastic rolling contact, shakedown isaccompanied with a significant reduction of the tangential loading capacity. This effectmust be considered in the design and construction of frictional contact systems underthe influence of vibrations of the type that is being regarded here.

39


3.3 Ratcheting of the Oscillating Rolling Contact

In case that fT or w exceed the shakedown limit stated in equation (3.2.18), i.e. fT >fT,lim or w > wlim, the contact fails and frictional ratcheting occurs. At first the systemresponses as in the shakedown case. The rolling causes alternating slipping processesat the actual trailing edge and the initial sticking area in the centre decreases. In turnthis increases the rigid body displacement within every period. As shown in Fig. 3.19the initial sticking area vanishes completely after only a few rolling periods. Thus, thesystem does not reach the shakedown equilibrium. Instead, one side of the contactalternately sticks, while the other slips in dependence on the actual rolling direction.This leads to an accumulated rigid body displacement of the substrate also referred toas ratcheting or walking [27, 28, 29] as shown in Fig. 3.4.

slip

stick +W −W

Ratcheting Ratcheting

+W

n = 0 n = 0.25 n = 2.6 n = 8.9

Fig. 3.19: distribution of stick (black) and slip (grey) area in case of ratcheting aftervarious numbers of rolling periods n for w = 0.24 and fT = 0.8 (CONTACT Simulation).Slipping occurs on the actual trailing edge while the actual leading edge sticks

The displacement per period, respectively the incremental displacement ∆u is an in-creasing function of fT and w, as depicted in Fig. 3.21. Here marks and error-barsdepict experimental results. Solid lines give an approximation function which is gainedon basis of the MDR model. For this purpose at first a parameter study is conductedusing the incremental MDR-simulation as described in appendix B.1. This shows that∆u is proportional to the supercritical portion of the oscillation amplitude:

∆u = η (w − wlim) . (3.3.1)

The constant of proportionality η is a function of ustat and is displayed in Fig. 3.20. Theblue line gives the numerical results on basis of the MDR simulation. The red line givesthe approximation function:

ηfit = 4.77√

ustat , (3.3.2)

which is the outcome of a linear regression analysis with ustat as the regressor. Forstatic displacements in the region of 0.2 ≤ ustat ≤ 0.8 this expression differs ±5%from the results of the MDR simulation. The resulting function is then adapted to theexperimental results what yields ηexp ≈ √

2ustat and thus:

∆u =√

2ustat (w − wlim) . (3.3.3)

40


The goodness of the fitting is given in appendix A.1.3 in Tab. A.1.1. The small deviationbetween ηfit and ηexp can again be explained with the violated rotational symmetry5,which is one of the basic assumptions of MDR. Finally, for consistency reasons, equa-tion (3.1.2) is inserted what gives the function that is depicted in Fig. 3.21:

∆u =

√

2(

1 − (1 − fT )2/3

)

(w − wlim) . (3.3.4)

It must be noted that wlim also depends on fT , as stated in equation (3.2.18). Theresults show qualitatively good agreement with those for the walking of a rocking punch,as examined by Mugadu et al. [27].

0.2 0.4 0.6 0.8

2

3

4

static displacement ustat

coeffi

cien

tη(u

sta

t)

ηηfit

Fig. 3.20: coefficient η as a function ofthe static displacement. Blue line givesresults of MDR simulation and red linegives approximation function (3.3.2)

0 0.2 0.4 0.6 0.8 10

0.5

1

tangential force fT

incr

.dis

pla

cem

ent

∆u w = 0.24

w = 0.51w = 0.74

Fig. 3.21: incremental displacement ∆uas a function of the tangential force fT fordifferent oscillation amplitudes w in caseof ratcheting. Solid lines give approxima-tion function (3.3.4)

Once the oscillating rolling stops, the incremental displacement stops as well. Thus, theratcheting effect must not only have a negative impact, but can also be used for thegeneration and control of small displacements in case that an increase of the tangentialloading is impossible or if high accuracy is needed as in micro-electromechanical systems(MEMS). Using equation (3.3.4) one can calculate this supercritical system response.

5In fact the contact region isn’t even axial symmetric for ratcheting of the oscillating sphere.

41


3.4 Influence of the Alignment of Load and Oscillation

In addition to the system introduced in section 3.1, also the case of oscillation andtangential loading being perpendicular to each other is examined. This setting is shownin Fig. 3.22 (b) where W⊥ describes the rolling amplitude of the centre of the sphere inthe y-direction.

µ = 0

xz

2a

FN

FT

FT

U

substrate

(a) static tangential contact

xy

FT

U

2W⊥

(b) perpendicular rolling

Fig. 3.22: (a) static tangential contact and (b) oscillating, elastic rolling contact with thelateral motion of the sphere W⊥ being perpendicular to the tangential loading FT

The normalization of the parallel setting given in equation (3.1.4) is used for the per-pendicular setting. Likewise, tangential forces below the maximum holding force andoscillation amplitudes smaller than the contact radius are assumed. Only numerical re-sults of the CONTACT simulation are reported here. For the simulation of a both casesa world fixed approach with 61 x 50 quadratic discretization elements with side length∆x, ∆y = 0.2 mm and the parameters of the experimental setting as described in section5.1.2 in Tab. 5.1.2 is used. The numerical model is described in appendix B.2.

0 2 4 6 8 10

0.35

0.4

0.45

ustat

usd

periods n

dis

pla

cem

ent

u

‖⊥

Fig. 3.23: displacement of the substrateu for w ‖ fT and w ⊥ fT with w = 0.23and fT = 0.46. Displacement stops andshakedown occurs

0 2 4 6 8 10

0.5

1

1.5

2

ustat

periods n

dis

pla

cem

ent

u

‖⊥

Fig. 3.24: displacement of the substrateu for w ‖ fT and w ⊥ fT with w = 0.23and fT = 0.77. Displacement continuesand ratcheting occurs

The system response as a result to the oscillation is the same in both cases, parallel and

42


perpendicular. Again, it shows that the oscillatory rolling initially leads to an increase ofthe rigid body displacement with respect to the static value, as depicted in Fig. 3.23 andFig. 3.24. In case that fT and w fall below the shakedown limits, the displacement stopsafter a certain number of periods and a shakedown occurs as in Fig. 3.23. The systemreaches a new equilibrium and the displacement refers to the constant, time independentshakedown displacement u = usd [25] as described in section 3.2. Otherwise, if fT and wexceed the shakedown limits, the contact fails and the displacement continues with theoscillation as in Fig. 3.24. An accumulated displacement described as ratcheting occurs[27] as described in section 3.3.


Whether the alignment is parallel or perpendicular, the system response to the rollingis the same. Alternating slip processes occur at the actual trailing edge that increasethe rigid body displacement. Finally, the system reaches a new equilibrium where threecharacteristic contact regions occur as depicted in Fig. 3.25. Here the traction τ alongthe centre line aligned parallel to the rolling is given for both settings, i.e. τ (x, y = 0) forw ‖ fT respectively τ (x = 0, y) for w ⊥ fT . The parallel centre lines are also symbolizedby the dashed lines in Fig. 3.29.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

stick-region

slip-region

zero-tract.

coordinate x/a (‖), y/a (⊥)

trac

tion

τ/µ

p 0

‖⊥

Fig. 3.25: traction τ/µp0 after shake-down along the centre line aligned paral-lel to the rolling for w ‖ fT and w ⊥ fT

with w = 0.23 and fT = 0.46 (CON-TACT model)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

stick-region

slip-region

coordinate y/a (‖), x/a (⊥)

trac

tion

τ/µ

p 0

‖⊥

Fig. 3.26: traction τ/µp0 after shake-down along the centre line aligned per-pendicular to the rolling for w ‖ fT andw ⊥ fT with w = 0.23 and fT = 0.46(CONTACT model)

It shows that the contact configuration is very much the same in both cases the onlydifference being the alignment, i.e. x for w ‖ fT and y for w ⊥ fT :

stick-region: Ux = const. ,slip-region: τ = µp ,

zero-traction: τ = 0 .(3.4.1)

43


Fig. 3.26 gives the traction aligned perpendicular to the rolling direction. The perpen-dicular centre lines are symbolized by the solid lines in Fig. 3.29. In both cases thezero-traction region almost vanishes along the perpendicular line. However, the trac-tion distribution slightly differs for both cases. Along the parallel centre line shown inFig. 3.25, it is higher for the perpendicular setting. And vice versa, along the perpen-dicular centre line, which is depicted in Fig. 3.26, it is higher for the parallel case. Thedifference of the displacement of particles is more obvious, as illustrated in Fig. 3.27 andFig. 3.28, which show ux along the centre lines. In accordance with the observations ofsection 3.4, ux in the stick region is higher for fT ⊥ w . However, along the parallel cen-tre line, ux in the slip- and zero traction-regions is higher for w ‖ fT . This is expected,since the overall tangential force is the same in both cases and a higher displacement inthe centre is equalized by a lower displacement in the outer region. The difference in thetraction and displacement distribution is caused by the evolution of the contact duringthe shakedown process. As a result, the direction of the oscillation influences the finalshakedown displacement.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

stick-region

slip-region

zero-

tract.

coordinate x/a (‖), y/a (⊥)

dis

pla

cem

ent

ux

‖⊥

Fig. 3.27: tangential displacement ux

after shakedown along the centre linealigned parallel to the rolling for w ‖ fT

and w ⊥ fT with w = 0.23 and fT = 0.46(CONTACT model)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

stick-region

slip-region

coordinate y/a (‖), x/a (⊥)

dis

pla

cem

ent

ux

‖⊥

Fig. 3.28: tangential displacement ux

after shakedown along the centre linealigned perpendicular to the rolling forw ‖ fT and w ⊥ fT with w = 0.23 andfT = 0.46 (CONTACT model)

3.4.2 Shakedown Limits for the Perpendicular Case

As in the parallel case, using the MDR the perpendicular rolling contact is mapped toa one-dimensional elastic foundation and a rigid indenter with stretched radius. Theinfluence of the oscillating rolling on the vertical displacement of springs in contact Uz

can again be directly deduced from kinematic relations. In the reversal points of therolling for both settings applies:

Uz (s) = d − (s ∓ W )2

Rfor − a ± W ≤ s ≤ a ± W . (3.4.2)

44


Here, the coordinate s points in the x-direction for w ‖ fT and in the y-direction forw ⊥ fT , as shown in Fig. 3.29. The influence of the rolling on the normal contact isthus identical in both cases6. However, since the MDR requires rotational symmetryand decoupled systems, the tangential displacements and the contact region must be thesame for both settings7. Consequently, the equilibrium condition is the same for bothsettings:

fT = 1 − 3

2wusd − (1 − usd)

3/2 . (3.4.3)

However, a different relation between loading and oscillation amplitude, i.e. a differentmapping parameter κ, is expected:

w ‖ fT ⇒ fT = 1 − κ‖wusd − (1 − usd)3/2 , (3.4.4)

w ⊥ fT ⇒ fT = 1 − κ⊥wusd − (1 − usd)3/2 . (3.4.5)

Comparison with the simulation and the experiments yields the different mapping pa-rameters for the two cases:

κ‖ = 1 and κ⊥ = 1.1 . (3.4.6)

what gives:

w ‖ fT ⇒ fT = 1 − wusd − (1 − usd)3/2 , (3.4.7)

w ⊥ fT ⇒ fT = 1 − 1.1wusd − (1 − usd)3/2 . (3.4.8)

Again, the goodness of the fitting is shown in the appendix A.1.3 in Tab. A.1.1.

xxyy zz

ss

R1D

R1D

FTFT

UU

2W‖

2W⊥

Fig. 3.29: equivalent one-dimensional MDR model. The control variable s is aligned in thex-direction for the parallel case (left) and aligned in the y-direction for the perpendicularcase (right)

Fig. 3.30 shows the shakedown displacement usd as a function of the tangential force fT

for different amplitudes w. Dashed lines are computed with equation (3.4.7) and solidlines are computed with equation (3.4.8). The error-bars and marks show experimentalresults for w ⊥ fT . Experimental results for w ‖ fT are given in section 3.2. The

6The reason for the change of vertical displacement, i.e. parallel or perpendicular rolling, does notmake a difference for one specific spring.

7At least within in the MDR model.

45


corresponding CONTACT simulation results are indicated by the asterisks. As for theparallel case, values close to the shakedown limit are not given because CONTACT lacksaccuracy close to the traction bound [88]. Comparison of the theoretical, experimentaland numerical results shows that the shakedown displacement usd is slightly higher fora perpendicular setting of tangential load and rolling direction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

usd − ustat

ustat

ratcheting

tangential force fT

shak

edow

ndis

pla

cem

ent

usd

w = 0w = 0.23w = 0.48w = 0.71ulim

Fig. 3.30: shakedown displacement usd as a function of fT for different w. Solid lines give(3.4.8) and dashed lines give (3.4.7). The oscillating rolling increases the displacement incomparison to the static value ustat. Perpendicular alignment w ⊥ fT leads to higher usd

than parallel alignment w ‖ fT

The dash-dot line gives the maximal displacement ulim as a function of fT , that isachieved before complete sliding occurs and the contact fails. With the vanishing stickradius condition csd = 0 and equation (3.2.10) one gets the same expression for the twocases:

w ‖ fT ⇒ ulim = 1 − w2T,lim , (3.4.9)

w ⊥ fT ⇒ ulim = 1 − w2T,lim . (3.4.10)

However, the relations between maximal tangential load and amplitude differ slightly.With ulim and equation (3.4.7) and equation (3.4.8), the maximal tangential forces read:

w ‖ fT ⇒ fT = fT,lim = 1 − wlim , (3.4.11)

w ⊥ fT ⇒ fT = fT,lim = 1 − 1.1wlim + 0.1w3lim . (3.4.12)

The shakedown limits are depicted in Fig. 3.31. Experimentally, the maxima are identi-fied by a stepwise increase of the amplitude, while the tangential force is kept constant,

46


see also section 5.1.4. The first amplitude for which the contact fails is identified as theshakedown limit. Thus, Fig. 3.31 shows wlim as a function of fT,lim. Again, experimen-tal results are illustrated with triangles and error-bars and are only given for w ⊥ fT .Experimental results for w ‖ fT are given in section 3.2. The theoretical values arerepresented by the blue line for w ‖ fT and by the red line for w ⊥ fT . It shows thatwlim is slightly lower for w ⊥ fT . However, the difference is in the range of the relativedeviation. Additionally, Fig. 3.32 shows the maximum shakedown displacement ulim asa function of fT,lim. Experimental values are again only given for w ⊥ fT , see section 3.2for results of the parallel case. The experiment shows relatively good agreement withthe theory, which is indicated by the red line for w ⊥ fT respectively the blue line forw ‖ fT .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


amplitu

de

wli

m

‖⊥

Fig. 3.31: maximal amplitude wlim as afunction of the maximal tangential forcefT,lim for w ‖ fT and w ⊥ fT . Trianglesgive the experimental results for w ⊥ fT ,solid lines gives the theoretical values of(3.4.11) and (3.4.12)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


dis

pla

cem

ent

uli

m

‖⊥


as a function of the maximal tangentialforce fT,lim for w ‖ fT and w ⊥ fT .Circles give the experimental results forw ⊥ fT , solid lines gives the theoreticalvalues of (3.4.9) and (3.4.10)

With the shakedown limits defined in equations (3.4.11) and (3.4.12), one can computethe maximal tangential force to achieve a safe shakedown for a given rolling amplitudeand vice versa. In addition, one can determine the corresponding maximal displace-ment. The results show that shakedown is accompanied with a significant reduction ofthe loading capacity, as fT,lim ≤ 1 applies in both cases. Additionally, for the sameamplitude, the parallel setting could theoretically bear slightly higher tangential load,but the difference is only minor. Thus, in force-locked connections with oscillations ofthis type, it might be more convenient if tangential loading and oscillation are alignedparallel.

3.4.3 Ratcheting for the Perpendicular Case

Once the shakedown limits are exceeded, the alternating slipping processes at the trailingedge will be continued with the rolling. In turn, the system does not reach a new

47


equilibrium and the contact fails. As shown in section 3.3, ratcheting occurs, whereone side of the contact slips, while the other sticks. This accumulated slip leads to arigid body motion [27]. Both, experiments and simulations show that the incrementalslip ∆u, i.e. the displacement per period, is an increasing function of fT and w. Asthe difference for the two settings, parallel and perpendicular, is only minor, the sameapproximation function as in section 3.3 is used. The incremental displacement ∆u forw > wlim is given as:

∆u = η (w − wlim) . (3.4.13)

Again, it should be emphasized that wlim also depends on fT . The constant of propor-tionality η is a function of fT , as shown in section 3.3. Adaption to the experimentalresults finally gives:

w ‖ fT ⇒ ∆u =

√

2(

1 − (1 − fT )2/3

)

(w − wlim) , (3.4.14)

w ⊥ fT ⇒ ∆u =

√

5(

1 − (1 − fT )2/3

)

(w − wlim) , (3.4.15)

where the goodness of the fits is shown in appendix A.1.3 in Tab. A.1.1. It turns out thatthe incremental displacement ∆u is significantly higher for the perpendicular setting ascould be seen in Fig. 3.33. Here, experimental results are symbolized by marks anderror-bars and are only given for w ⊥ fT . Dotted lines give equation (3.4.14) and thesolid lines give equation (3.4.15). The ratcheting effect can be used for the generationof small displacement in case that an increase of the tangential loading is not possibleor high accuracy is needed as in MEMS devices. The perpendicular setting is moreconvenient as for the same rolling amplitude and tangential force it generates a higherdisplacement.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

tangential force fT

incr

emen

tal

dis

pla

cem

ent

∆u

w = 0.23w = 0.48w = 0.71

Fig. 3.33: incremental displacement ∆u as a function of the tangential force fT for differentoscillation amplitudes w in case of ratcheting for w ‖ fT and w ⊥ fT . Marks and error-barsgive experimental results for w ⊥ fT . Dotted lines give the theoretical values of (3.4.14)and solid lines those of (3.4.15)

48


3.5 Oscillating Cylindrical Rolling Contact

One important extension of the oscillating rolling contact are cylindrical systems, sincesuch are often used in technical applications. Consider a tangentially loaded contact ofa rigid cylindrical body of length L and an elastic substrate, as sketched in Fig. 3.34(a). As before, the radius of the cylinder R and the elastic moduli E∗ and G∗ of thesubstrate, i.e. of the half-space, are effective quantities of two contacting bodies. Again,dry friction with constant µ, uncoupling displacements and a constant load regime FN

and FT are assumed. According to the Hertz theory [52] a pressure distribution8 isconsidered that is elliptic in the x, z-plane and constant in the y-direction:

p (x) = p0

(

1 −(

x

a

)2)1/2

for − a ≤ x ≤ a . (3.5.1)

Here p0 denotes the maximum pressure in the centre and a is the half width of thecontact:

p0 =2FN

πaLand a =

√

4RFN

πE∗L. (3.5.2)

With p (x) as in equation (3.5.1) the indentation depth of the cylinder d reads [89, 90]:

d =2FN

πE∗L

(

1

2(1 + ln (4)) + ln

(

L

a

))

. (3.5.3)

FN /L

FT

FT

µ, E∗, G∗

yx

z

L

d

2a

R

p(x)

(a) cylindrical contact

p0

2a

2c

x

p, τ

p(x) - pressure

τ(x) - traction

(b) pressure and traction

Fig. 3.34: (a) tangentially loaded contact of a rigid cylinder with radius R and length Land elastic half-space. (b) tangential contact with pressure p (x) and traction distributionτ (x). Incipient sliding: contact width 2a and stick region with width 2c in the centre

The tangential problem was derived independently by Cattaneo [59] and Mindlin [60]. Agood description of their procedure is given in [55] or [56]. Just as in the spherical case,

8In fact, the shape of the cylinder must rather slightly resemble to a barrel in order to achieve such apressure distribution in a real system [89]. However, such a profile is very difficult to manufacture andcorrect only at the design load [52].

49


which is briefly described in section 2.1.1 and section 2.1.2, a tangential force insufficientto cause complete sliding, i.e. FT ≤ FT,max = µFN , will lead to slipping only at theboundary region of the contact area. The centre region −c ≤ x ≤ c remains in a stateof stick, where the half stick width reads:

c = a

(

1 − FT

µFN

)1/2

. (3.5.4)

To maintain this condition known as incipient sliding, the traction in the contact τ (x)must be a superposition of two elliptical traction distributions of the Hertzian type asin equation (3.5.1). In the complete contact region −a ≤ x ≤ a applies the traction:

τ1 (x) = µp0

(

1 −(

x

a

)2)1/2

, (3.5.5)

while in the sticking region −c ≤ x ≤ c a second term is added:

τ2 (x) = −τ0

(

1 −(

x

c

)2)1/2

. (3.5.6)

The so called correctional term τ0 reads:

τ0 = µp0c

a(3.5.7)

and is given by the condition that the tangential displacement of particles must beconstant in the sticking region. Finally the traction in the static case is distributed assketched in Fig. 3.34 (b):

τ (x) = τ1 (x) + τ2 (x) for 0 ≤ |x| ≤ c ,τ (x) = τ1 (x) for c < |x| ≤ a .

(3.5.8)

However, no exact solution is known for the tangential displacement of the substrateUstat, one exception being elliptical contacts, where the displacements are expressed interms of elliptical integrals [64, 91]. Even if the stress distribution is constant in the y-direction, the problem cannot be treated as being two-dimensional due to the followingreason. In case of a two dimensional load, the displacement is decreasing logarithmicallywith r being the distance from the loaded surface [56]:

Ux ∝ −kln

(

r

r0

)

, (3.5.9)

where k is the constant of proportionality. The constant r0 is given by the conditionthat the displacement vanishes far away from the contact. Thus, the displacement in thecentre of the contact depends on the choice r0 and approaches infinity9 for r0 → ∞. It is

9Prescott has likened the situation to a load applied to an infinitely long rod fixed at one end. Theextension of such a rod caused by the application of any load would be infinite [90].

50


therefore necessary to treat the problem as fully three dimensional, even if the stress isconstant in the y-direction. Again, the tangential displacement for a given traction τ (x)and contact area is in principle determined by the potential functions of Boussinesq andCerruti [56]:

Ustat = Ux (0, 0, 0) =1

πG

L2∫

− L2

c∫

0

(

1 − ν

s+

νξ2

s3

)

(τ1 (ξ, η) + τ2 (ξ, η)) dξdη

+1

πG

L2∫

− L2

a∫

c

(

1 − ν

s+

νξ2

s3

)

τ1 (ξ, η) dξdη . (3.5.10)

with G and ν as equation (2.1.12) and s as in equation (2.1.5). Using equation (3.5.2)and (3.5.4) and the traction as in equations (3.5.5)-(3.5.8) one can conduct a numericalintegration10 of equation (3.5.10). Finally, curve fitting gives an approximation for thetangential displacement:

Ustat = U0

(

1 −(

1 − FT

µFN

))0.92

. (3.5.11)

Here, U0 denotes the maximum displacement for FT = µFN , which is also an approxi-mation based on the indentation depth of equation (3.5.3):

U0 = µ2FN

πE∗L

(

1

2(1 + ln (4)) + ln

(

1.8E∗

G∗L

a

))

. (3.5.12)

Again, the goodness of the fits is given in appendix A.1.3 in Tab. A.1.1.

3.5.1 Influence of the Oscillating Rolling

As for the spherical case, the static tangential contact is superposed by a slight oscillatoryrolling of the cylinder. Again, the amplitude W denotes the lateral movement of thecentre of the cylinder, as depicted in Fig. 3.35. Likewise, the overall macroscopic normaland tangential forces are kept constant in magnitude and the rolling corresponds to arocking of the contacting bodies. The parameters of influence and the displacement arenormalized as:

fT =FT

µFN, w =

W

a, u =

U

U0, (3.5.13)

where a refers to equation (3.5.2) and U0 is the maximal static displacement as statedin equation (3.5.12). Again, tangential forces below the maximum holding force andoscillation amplitudes smaller than the half contact width are assumed:

fT ≤ 1 and w ≤ 1 . (3.5.14)

10Even as both, integration limits and integrand, are known and several advanced symbolic mathe-matical software was used, it was unfortunately not possible to find a closed form solution.

51


Thus, taken by itself, neither of the two factors leads to a failure of the contact, exactlyas in the spherical case.

FN /L

FT

FT

µ, E∗, G∗

yx

z

L2W

Fig. 3.35: oscillating rolling contact of a rigid cylinder and an elastic half-space. Thecentre of the cylinder rolls with amplitude W and the macroscopic forces are constant.This corresponds to a rocking of the of the upper body

0 2 4 6 8 100.06

0.07

0.08

0.09

1.0

ustat

usd

periods n

dis

pla

cem

ent

u

w = 0.26w = 0.52w = 0.77

Fig. 3.36: displacement u for differentoscillation amplitudes w and fT = 0.08.Shakedown: displacement stops and sys-tem reaches a new equilibrium

0 2 4 6 8 100.5

1

1.5

2

2.5

ustat

periods n

dis

pla

cem

ent

uw = 0.26w = 0.52w = 0.77

Fig. 3.37: displacement u for differentoscillation amplitudes w and fT = 0.58.Ratcheting: displacement continues

Figures 3.36 and 3.37 give the displacement of the substrate u for different amplitudesw. It is computed using the CONTACT model, see also appendix B.2. The parametersof the experimental rig as in section 5.1.3 in Tab. 5.1.3 are used. The displacementcorresponds to the rigid body displacement between remote points within the cylinderand the substrate as the sum of the elastic displacement and the accumulated shift pertime step. As one expects, the system response is equivalent to the oscillating contactwith spherical rolling body. In combination with the constant macroscopic load, therolling leads to an increased rigid body displacement in relation to the static value.Figure 3.36 shows that the displacement stops after a few periods and the contact holds,even if the rolling continues. The system reaches a new equilibrium and the accordingdisplacement corresponds to the constant time independent shakedown displacement usd

52


[25]. For this effect to occur, fT and w must fall below the shakedown limits, otherwisethe contact fails. As for the sphere, the accumulated shift per time step leads to acontinuing displacement, as depicted in 3.37. This is referred to as frictional ratchetingor walking [27, 28, 29].


The system starts in equilibrium and the entire contact sticks. When the cylinder startsrolling, the pressure drops at the actual trailing edge and increases at the leading edge.In the reversal points the pressure is distributed as in Fig. 3.38 (a):

p (x) = p0

(

1 −(

x ∓ W

a

)2)1/2

for − a ± W ≤ x ≤ a ± W . (3.5.15)

This causes slipping at the trailing edge, while at the leading edge the traction initiallyremains constant. The resulting imbalance in the tangential direction increases therigid body displacement between cylinder and substrate u within every back and forthmovement. For sufficiently small fT and w, shakedown occurs and a saturation level isreached after a certain number of periods, as shown in Fig. 3.36.

+W −W

WW

2a2a

xx

p(x)p(x)p0p0

00

(a) pressure distribution

stick-regionslip-region

zero-traction

2a

2b

2csd

x

τ (x)

τ1−sd (x)

τ2−sd (x)

(b) traction distribution

Fig. 3.38: (a) pressure distribution p (x) and contact region in the reversal points, causedby the oscillatory rolling. (b) traction distribution τ (x) and contact regions (stick-region,slip-region, zero-traction) after shakedown

In the new equilibrium, there remain three characteristic contact widths, as depicted inFig. 3.38 (b). Firstly, in the centre region, the displacement must be constant because ofthe sticking condition (stick region). Secondly, adjacent thereto occurs a region, wherethe tangential stress equals the traction bound (slip region). Thirdly, at the outside ofthe contact, the cylinder is periodically released and the tangential traction is zero (zerotraction). In summary this gives:

stick-region: 0 ≤ |x| ≤ csd ⇒ Ux = const. ,slip-region: csd < |x| ≤ b ⇒ τ = µp ,

zero-traction: b < |x| ≤ a ⇒ τ = 0 .(3.5.16)

53


Here csd denotes the half stick width and for b applies b = a − W . In order to deducethe traction distribution, the slip region is considered firstly. In this region the tractionτ (x) must equal the friction coefficient times the minimal pressure distribution, thatappears at the reversal points of the oscillation, as stated in equation (3.5.15):

τ1−sd (x) = µp0

(

1 −(

x + W

a

)2)1/2

for csd < |x| ≤ b . (3.5.17)

In contrast to the static case, the traction in the centre region will be from a differenttype as this region is neither released nor slipping at any time. However, one can proceedin the Cattaneo [59] and Mindlin [60] manner and can propose that the traction in thestick region is again a superposition of two Hertzian distributions:

τ2−sd (x) = τ21

(

1 −(

x

b

)2)1/2

− τ22

(

1 −(

x

csd

)2)1/2

for 0 ≤ |x| ≤ csd . (3.5.18)

Each of these tractions causes a parabolic tangential displacement [92] of particles in thecentre region. The total displacement is a superposition of the individual displacements:

Ux (x) = C1 − τ21

bE∗ x2 + C2 +τ22

csdE∗ x2 (3.5.19)

that must be constant in the stick region in accordance to the no slip condition. Figure3.39 gives the displacement Ux in the entire contact. It is computed using the CONTACTmodel. The sticking condition yields:

εxx (x) =∂Ux

∂x= 0 ⇒ τ22 = τ21

csd

b. (3.5.20)

Further, the traction must be continuous at the edge of the sticking region x = csd. Withequations (3.5.17), (3.5.18) and (3.5.20) this gives

τ1−sd (x = csd) = τ2−sd (x = csd) ⇒ τ21 =µp0

(

1 −(

csd+Wa

)2)1/2

(

1 −( csd

b

)2)1/2

. (3.5.21)

Finally, the complete traction τ (x) after shakedown is distributed as in Fig. 3.38 (b):

stick-region: 0 ≤ x ≤ csd ⇒ τ (x) = τ2−sd (x) ,slip-region: csd < x ≤ b ⇒ τ (x) = τ1−sd (x) ,

zero-traction: b < x ≤ a ⇒ τ (x) = 0 .(3.5.22)

It must be noted that csd is still unknown. After shakedown, the integral of the tractionover the contact width Lc = 2a must match the tangential loading per length:

FT

L=

∫

Lc

τ (x)dx = 2

csd∫

0

τ2−sd (x)dx + 2

b∫

csd

τ1−sd (x)dx . (3.5.23)

54


With equations (3.5.17), (3.5.18), (3.5.20) and (3.5.21) this gives:

fT =2

πarccos

(

csd

a+ w

)

+2

π

(

1 −(

csd

a+ w

)2)12

b

a

arcsin( csd

b

)

−( csd

b

)2 π2

(

1 −( csd

b

)2)12

− w

. (3.5.24)

Using numerical solving procedures this transcendent equation gives the unknown stickradius csd for different fT and w. The results are depicted in Fig. 3.40. For a mutualverification csd is also computed using the CONTACT model. Lines depict the resultsusing equation (3.5.24) and marks depict the results gained via simulation. Both solu-tions show strong agreement. As one expects, the half stick-width decreases if fT and ware increased.

−1 −0.5 0 0.5 1

0.2

0.4

0.6

fT = 0.22

fT = 0.44

fT = 0.66

width x/a

dis

pla

cem

ent

ux

Fig. 3.39: tangential displacement ofparticles ux for different fT and w = 0.26computed with CONTACT model. Dis-placement is constant in the centre

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

tangential force fT

hal

fst

ick

wid

thc s

d/a w = 0.26

w = 0.52w = 0.77

Fig. 3.40: half stick width csd aftershakedown as a function of fT for dif-ferent w computed with (3.5.24) (solidlines) and CONTACT model (marks)

With these results, it is now possible to compute the missing traction constants τ21 andτ22 and thus τ (x). Again, the simulation is used for a mutual verification. The tractiondistribution for different fT is shown in Fig. 3.41. Here, the dashed lines show the CON-TACT solution, whereas the analytical expression is denoted by the solid lines and justgiven for one half of the symmetrical contact. Both solutions show strong agreement. Asone can see, the stick region decreases and the maximal traction increases with fT . Sincew is not changed, the slip radius b is constant for all forces considered. Figure 3.42 de-picts the traction for different amplitudes w. Higher amplitudes decrease the stick regionand increase the periodically released area, i.e. the zero traction region. Additionally,the magnitude of the remaining traction is amplified. This is because the remainingarea with non-zero traction decreases at a constant tangential force. This illustrates theshakedown effect, where the residual force in the contact must be sufficiently strong towithstand the tangential load and to prevent any further sliding [25].

55


−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

2b

2csd

fT = 0.22fT = 0.44fT = 0.66

width x/a

trac

tion

τ(x

)/µ

p 0

Fig. 3.41: normalized tangential trac-tion distribution τ (x) after shakedownfor w = 0.26 and different fT

−1 −0.5 0 0.5 10

0.2

0.4

0.6

2b

2csd

w = 0.26w = 0.52w = 0.77

width x/a

trac

tion

τ(x

)/µ

p 0

Fig. 3.42: normalized tangential trac-tion distribution τ (x) after shakedownfor fT = 0.08 and different w

3.5.3 Shakedown of the Cylindrical Rolling Contact

With the traction after shakedown being known, it is possible to compute the accordingtangential displacement of the substrate usd. Again, the potential functions of Boussinesqand Cerruti for symmetric τ (x) [56] are used:

Usd = Ux (0, 0, 0) =1

πG

L2∫

− L2

csd∫

0

(

1 − ν

s+

νξ2

s3

)

τ2−sd (ξ, η) dξdη

+1

πG

L2∫

− L2

b∫

csd

(

1 − ν

s+

νξ2

s3

)

τ1−sd (ξ, η) dξdη , (3.5.25)

where G and ν and s are as in equation (2.1.12). However, since the exact solutionfor the half stick width csd remains unknown, it is not possible to give a closed-formsolution for equation (3.5.25). Instead, equation (3.5.24) is used to compute csd fordifferent fT and w. Subsequently, the results are inserted into equation (3.5.25) andnumerical integration procedures are used to get Usd. Finally, curve fitting tools areused to compute an approximation for the normalized displacement as a function of fT

and w:

usd = 1 − (1 − fT )0.92 + 0.3fT w . (3.5.26)

The goodness of the fit is given in appendix A.1.3 in Tab. A.1.1. The shakedown dis-placement for different fT and w is given in Fig. 3.43. Solid lines give the results usingequation (3.5.26) and asterisks give the CONTACT simulation results. Error-bars andmarks indicate the experimental results. As for the sphere usd is increased in comparison

56


to the according static value ustat. It shows that the bandwidth of the displacements isnarrower than for the spherical contacts.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

usd − ustat

ustat

ratcheting

tangential force fT

shak

edow

ndis

pla

cem

ent

usd

w = 0w = 0.28w = 0.52w = 0.76ulim

Fig. 3.43: shakedown displacement usd as a function of the tangential force fT for differentoscillation amplitudes w in case of a cylindrical roller. Solid lines give theoretical result(3.5.26). The oscillatory rolling increases the displacement by comparison with its staticvalue ustat

The dash-dot line shows the maximal displacement ulim that is reached before completesliding occurs and the contact fails. If so, the stick width csd goes to zero. In combinationwith equation (3.5.24) this gives the exact analytical relation between maximal tangentialload fT,lim and maximal amplitude wlim:

fT,lim =2

π

(

arccos (wlim) − wlim

(

1 − (wlim)2)1/2

)

. (3.5.27)

Curve fitting again gives approximations for the inverse relation:

wlim = 1 − f0.73T,lim , (3.5.28)

as well as for the maximum displacement, see appendix A.1.3 and Tab. A.1.1:

ulim = f0.87T,lim . (3.5.29)

The equations (3.5.27) and (3.5.28) enable to specify the highest possible amplitude tomaintain a safe shakedown for a given tangential force and vice versa. Again, the CON-TACT model is used for verification. Starting points are rather small amplitudes thatare increased stepwise until ratcheting occurs. Fig. 3.44 depicts the maximum amplitude

57


as a function of the tangential force. Equation (3.5.28) is symbolized by the solid line.Triangles give the simulation results, where upward pointing ones depict the last shake-down amplitude (sd) and downward pointing ones depict the first ratcheting amplitude(rt). Also experimentally, the maxima are identified by a stepwise increase of w, seesection 5.1.4 for a description of the experimental procedures. The results are shown byasterisks (exp) and error-bars. The deviations between equation (3.5.28) and the exper-iments are relatively high. The good agreement with the numerical values indicates afaulty experiment. One explanation for this might be a deviation of the actual pressuredistribution from the assumed cylindrical one, as discussed in section 5.1.5. Besides, thedetermination of the maximal amplitude is not straight forward but relatively artificialas described in section 5.1.4. Figure 3.45 shows the according maximum displacements.Formula (3.5.29) (solid line), simulation (triangles) and experiment (asterisks and error-bars) are in good agreement.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


amplitu

de

wli

m

sdrtexp

Fig. 3.44: maximal amplitude wlim asa function of the tangential force fT,lim.Solid lines give (3.5.28)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


dis

pla

cem

ent

uli

m

sdexp


as a function of the tangential forcefT,lim. Solid lines give (3.5.29)

3.5.4 Ratcheting of the Cylindrical Rolling Contact

As for the sphere, ratcheting occurs if the shakedown limits stated in equations (3.5.27)and (3.5.28) are exceeded. Slip occurs on the actual trailing edge of the contact thataccumulates to a continuing rigid body motion. The incremental slip per period ∆u fordifferent fT and w is given in Fig. 3.46. The solid lines show the approximation function:

∆u = 0.64fT (w − wlim) , (3.5.30)

which is computed using a linear regression wtih ustat being the regressor, see detailsin appendix A.1.3 in Tab. A.1.1. For this purpose the CONTACT model is used toincrease the number of data points. The obtained coefficient is then adjusted to theexperimental data. Except for low amplitudes w experiment and equation (3.5.30) arein good agreement. It shows that ∆u is lower for the cylindrical contact than for thespherical ones, where the incremental displacement is given in Fig. 3.33.

58


0 0.2 0.4 0.6 0.8 10

0.2

0.4

tangential force fT

incr

emen

tal

dis

pla

cem

ent

∆u

w = 0.28w = 0.52w = 0.76

Fig. 3.46: incremental displacement ∆u as a function of the tangential force fT for differentoscillation amplitudes w in case of ratcheting. Marks and error-bars give experimentalresults, solid lines gives the theoretical value of (3.5.30)

3.6 Summary

As generic models for force-locked connections under the influence of vibrations, oscil-lating rolling contacts between spherical and cylindrical rollers and a flat substrate havebeen introduced. Basic assumptions are a constant load regime FT and FN , dry frictionof the Coulomb type with a constant µ and linear elastic material behaviour. Addi-tionally, the systems are assumed to be quasi-static and from an uncoupled type, i.e.Dundur’s constant β = 0, meaning that variations in the normal force will not induceany tangential displacement and vice versa.It shows that slight oscillatory rolling of the roller varies the pressure distribution andthe contact region within every cycle. In turn, this leads to partial slip and macroscopicrigid body displacement of the substrate. Depending on both, oscillation amplitude wand tangential loading fT,lim, the displacement stops after a few periods or continues.The former case is referred to as shakedown and the latter as ratcheting. The exactshakedown limits for both, oscillation amplitude and tangential load for the three casesconsidered have been derived: spherical roller with parallel and perpendicular alignmentof load and oscillation and cylindrical roller with parallel alignment. In all three cases itturns out that shakedown is accompanied with a significant reduction of the tangentialload capacity, which is approximately:

fT,lim ≈ 1 − wlim . (3.6.1)

In addition, using the results one can predict the rigid body motion for the shakedowncase and the incremental slip in case of ratcheting. The comparison of experiment andtheory shows that the method of dimensionality reduction (MDR) has proven to be asuitable instrument for the modelling of oscillating rolling contacts.

59


60

Chapter 4

Dynamic Influences on Sliding

Friction

System dynamics is a crucial factor for the macroscopic frictional resistance of dynamicsystems [5, 9, 45]. In the following chapter an explanation for the underlying effectsis proposed and a theoretical model is introduced that captures the basic mechanisms.A non-dimensional form of the governing equations of motion is derived and the non-dimensional parameters are identified. In the next step numerical simulation is usedto determine the parameter range that enables to induce a significant reduction of thefrictional resistance. Finally, the theoretical outcomes are compared with experimen-tal results. Taken together, it shows that characteristic self-excited oscillations withvibrations in the normal direction can lead to an almost vanishing frictional resistance.

4.1 The Micro-Walking Machine

A potential basic system that captures important dynamic effects that enable a signif-icant reduction of the apparent frictional resistance could be as follows. Assume anelastic specimen which is pressed on a rigid substrate by the macroscopic normal forceFN . In the tangential directional direction it is additionally loaded with the macroscopictangential force FT , such that it slides with a constant velocity v0 like in Fig. 4.1 (a).

FN

FTv0

µ

x

z

(a) elastic specimen

FN

FTv0

FNiFT i

(b) free body diagram

Fig. 4.1: basic model for the micro-walking effect. (a) elastic specimen moving on a rigidsubstrate with constant velocity v0. (b) free body diagram in rotated position with contactforces of contact spot i

61

Chapter 4. Dynamic Influences on Sliding Friction

As in real contacts, specimen and substrate will not touch over the entire apparentcontact surface but in several small contact zones [3]. Each contact zone will exhibitits own stress distribution in the normal and tangential direction which is representedby a normal and a tangential force FNi and FT i as depicted in Fig. 4.1 (b). Only thecase of dry friction is considered and it is assumed that Coulomb’s law with a constantcoefficient µs = µk = µ applies in the contact spots. Thus, any distinction between staticand kinetic friction and the variation of the latter with sliding speed are neglected, asdiscussed in section 2.3.3. The coefficient µ reflects all microscopic influences of thesurface that depend on the material pairing. According to the local sticking condition aspecific spot sticks, whenever:

|FT i| ≤ µFNi . (4.1.1)

Otherwise, the contact spot slips and is subjected to a local resistance force:

FRi = µFNi . (4.1.2)

Due to the kinematic coupling between the normal and tangential translation and therotation of the rigid body, the forces in the contact spots will vary in time. This effectsignificantly influences the apparent frictional resistance [10, 45] as will be explained inthe following. Assuming that there are overall n similarly loaded contact spots in thesystem, one can define the theoretical average normal force acting in one contact spot:

FNi =FN

n. (4.1.3)

Due to the elasticity of the specimen, the actual status of a specific contact spot, i.e.sticking or slipping, is independent from the actual status of the rigid body. Thus, therigid body can move in the tangential direction while a single spot is sticking. In addition,the distance between the spots is assumed to be sufficiently large so that the spots areindependent. The relative displacement between substrate and bulk body therefore onlychanges the elastic deformation in the vicinity of a sticking spot and a slipping spotsimply moves along with the bulk body. Under these assumptions the system will beable to micro-walk on the substrate. In this case micro-walking describes a particularbehaviour of the contact spots:

• a contact spot only slips whenever its actual normal force FNi falls below FNi

• a contact spot sticks whenever its actual normal force FNi exceeds FNi

As a result, the average resistance force of contact spots FRi = 〈FRi〉, where 〈.〉 denotesthe time average, falls below the expected value calculated with (4.1.2) and (4.1.3):

FRi < µFNi . (4.1.4)

In consequence, the overall resistance force of the system, which is the sum over all ofthe n contact spots, will fall below the overall theoretical value as well:

FR =n∑

i=1

FRi <n∑

i=1

µFNi = µFN . (4.1.5)

62


This leads to the definition of the effective coefficient of friction µe:

µe =FR

µFN≤ 1 . (4.1.6)

There is a strong analogy to the ratcheting case described in section 3.3, where one sideof a single contact slips while the other sticks [27]. The micro-slip accumulates to a rigidbody motion. Hence, ratcheting decreases the tangential load that is needed for grosssliding, i.e. decreases the effective static and kinetic friction.An experimental analogy of the dynamic mechanism of friction reduction is given bythe work of Tolstoi [39]. In this case, increasing sliding speed increases the normaloscillations of the sliding body. Due to the non-linearity between indentation and normalforce, see section 2.1.1, the vibrations of the slider are highly asymmetric. Consequently,an increase of the normal oscillation amplitude decreases the mean value of indentationand the contact radius of the contact spots during sliding. As a result, the friction forcedecreases as well [93, 10]. The micro-walking machine extends this concept, as it alsoincludes the spatial variation of normal and tangential forces and stick and slip zones.

4.1.1 Discrete Model

The simplest implementation of a system that captures the aforementioned effects is aplane specimen that has only two contact spots at the edges, as depicted in Fig. 4.2 (a).The specimen is unloaded in the normal direction except for the force of gravity mgof the bulk body. Here m denotes the mass of the specimen and g is the gravitationalacceleration. The tangential force FT is applied by a constantly moving base, that isconnected via a spring with stiffness ks. As the base moves with constant velocity v0,the mean velocity of the specimen v exactly corresponds to v0. This type of excitationmodels a nominally steady motion of the system. Thus, any vibrations that may occurare not directly caused by the excitation but rather by self-excited oscillations.The separation of space scales principle as is introduced in section 2.3.1 is used for afurther simplification. According to this, the space scales contributing to the elastic andkinetic energy of a mechanical system with friction can be separated in the followingway [65]:

• The elastic energy is a local quantity which only depends on the conditions in thecontact region

• The kinetic energy is a non-local quantity which can be assumed to equal to thekinetic energy of the rigid bulk as a whole

This enables a further simplification, such that there remains a rigid body with massm, moment of inertia θ, height 2a and length 2b that consists of two elastic contactspots as shown in Fig. 4.2 (b). These spots are modelled as linear springs with normalstiffness kz and tangential stiffness kx as described in section 2.3.3. The stiffness can beapproximated as [55]:

kz = E∗Dc and kx = G∗Dc . (4.1.7)

63


Here E∗ and G∗ denote the effective elastic modules, that are a function of the shearmodulus G and Poisson’s ratio ν of the elastic specimen [55]:

E∗ =2G

1 − νand G∗ =

4G

2 − ν. (4.1.8)

The contact diameter Dc depends on the instantaneous contact configuration and isestimated as shown in section 5.2.1. The deflections of the springs in the normal andtangential direction Uzi and Uxi depend on the actual state of motion of the rigid body,i.e. they are a function of x, z and ϕ. These deflections represent the elastic deformationsof the specimen in the normal and tangential direction. This can be illustrated as follows.Considering Fig. 4.3 (a) it shows that the change of position of a point on the rigid bodyP to P ′ causes deformations that are concentrated in the vicinity of the contact spots ina volume with linear dimensions of the order of the contact width Dc [65]. As shown inFig. 4.3 (b), in the theoretical model this effect is considered by the spring deflections.The system is also assumed to be decoupled as introduced in section 2.1.3. Thus, achange in Uz does not change Ux and vice versa1. Consequently, the contact forcesacting on the rigid body are given as:

FN1/2 = kzUz1/2 and FT 1/2 = kxUx1/2 , (4.1.9)

where the subscript (.)1/2 denotes the two contact spots as shown in Fig. 4.2 (b). Fi-nally, in the simplified model, there remain only three degrees of freedom: X and Z,which describe the lateral and the vertical translation of the centre of gravity of thespecimen, and Φ, which describes the rotation of the rigid body. In addition, there aretwo dependent variables for each contact spot: the spring deflections Uz1/2 and Ux1/2.Still, the moving base acts as the only external excitation, where h denotes its lever armwith respect to the centre of gravity. This configuration allows the lever arm to be largerthan the height a, as examined in section 4.2.3.

mg ks

v0

µ

G, ν

g h

D

(a) specimen

2b

2av0

µ

ks

kz, kx

1 2

g h

Φ

X

Z

mg

(b) simplified system

Fig. 4.2: (a) elastic specimen consisting of two spherical shaped edges pulled by a con-stantly moving base. (b) simplified rigid body model with three degrees of freedom andlinear springs to model the contact spots

1Nevertheless, the deflections are coupled through the equations of motion of the system.

64


G, νDc

PP ′

(a) contact spot

kz

kx

Ux

UzP

P ′

(b) linear spring

Fig. 4.3: (a) elastic contact spot, where deformations are concentrated in a volume withlinear dimensions of the order of the contact width Dc. (b) spring element with normalstiffness kz and tangential stiffness kx and corresponding deflections Uz and Ux. The dottedlines show the old positions of the rigid bodies

There is an analogy to the model proposed by Martins et al. which consists of a rigidblock that exhibits the same degrees of freedom, i.e. translation and rotation [10]. Inthe normal direction the rigid block is as well only subjected to its own weight. And inthe tangential direction it is restrained by a spring and a viscous damper (dash-pot) andsubjected to a moving belt. Also the interface friction is modelled assuming Coulombdry friction with contact stiffness in the normal and tangential direction. This enablesto compare the results. However, their stress distribution in the contact surface is adirectly determined function of the rigid body motion and does not enable parts of thecontact to slip, while others stick as in the model introduced here. Thus, on the onehand the present model is an extension of the model proposed by Martins et al. whichalso takes into account the spatial variation of stick and slip zones. On the other hand,the non-linearity of the interface response [10] is neglected.Again it should be emphasized that this work considers the known basic mechanisms thatare responsible for the experimentally observed dependency of the frictional resistanceon the system dynamics. These are essentially vibrations in the normal direction [39, 40,93, 43] and coupling effects between normal, tangential and rotational degrees of freedom[10, 44, 45]. In the present model, a rotation of the body Φ, i.e. a pitching motion, leadsto varying normal and tangential spring deflections Uz1/2 and Ux1/2 and therefore forcesat both contacts. The other way round, this asymmetry acts as a twisting momenton the rotation. Thus, an appropriate synchronization of these motions will enablethe system to walk in such a way as explained in the beginning of this chapter. Thequestions remain, under which conditions the excitation of the constantly moving basecan cause self-excited oscillations of the rigid body and whether these oscillations willbe synchronized in the proposed manner.

4.1.2 Parameters of Influence

The initial model is described by eleven dimensionful parameters. According to theso called Π-theorem [94] the number of parameters can be reduced introducing a non-dimensional representation of the governing equations. Therefore the displacements and

65


deflections are normalized with the geometrical constants a and b:

x =X

b, z =

Z

b, ϕ =

a

bΦ, uz1/2 =

Uz1/2

b, ux1/2 =

Ux1/2

b. (4.1.10)

In addition, the characteristic period τ and the dimensionless time t are introduced:

τ =

√

m

kx⇒ t =

tdim

τ, (4.1.11)

where tdim denotes the dimensionful time. This yields the accelerations as:

X =d2X

dt2dim

= bd2x

d (tτ)2 =kxb

m

d2x

dt2=

kxb

mx′′ , (4.1.12)

Z =kxb

mz′′ , (4.1.13)

Φ =kxb

maϕ′′ , (4.1.14)

where (.)′ denotes the derivation with respect to the non-dimensional time t. Thisprocedure reduces the number of parameters from eleven to eight, the remaining onesbeing listed in Tab. 4.1.1. Considering the analogy to the model of Martins et al. [10]the parameter κ1 corresponds to their stiffness parameter sM that takes into accountthe ratio of the stiffness of the base spring and the normal contact stiffness. In additionparameter κ3 is equivalent to their ratio of height and width hM . Finally, the microscopicfriction µ resembles to their more sophisticated friction parameter fM . Despite thedifferences between the two models this enables to compare the overall trend of theinfluences of the parameters in the following sections.

Tab. 4.1.1: definitions and interpretation of the remaining parameters of the non-dimensional system

parameter definition physical interpretation

κ1kskx

ratio of macro and micro stiffness

κ2kzkx

ratio of normal and tangential contact stiffness

κ3ab geometry ratio, slenderness of the rigid body

κ4mgkxb ratio of normal force and contact stiffness times half width

κ5 µ microscopic friction

κ6v0

b

√

mkx

velocity dependent non-dimensional parameter

κ7Θ

mb2 ratio of rotational inertia of the rigid body

κ8ha ratio of the lever arm of the excitation

66


4.1.3 Modelling and Simulation

Figure 4.4 depicts the free-body diagram of the micro-walking machine in a deflectedposition. In the non-dimensional form of the system, the deflections of the springscorrespond to the contact forces:

fN1/2 = κ2uz1/2 and fT 1/2 = ux1/2 , (4.1.15)

where the subscript (.)1/2 again denotes the two contact spots. Depending on the instan-taneous motion, the contact spots might be in contact or be released from the substrateas shown Fig. 4.4. Here, the left contact (1) is fully released, what will lead to zerocontact forces. In case that a spring is in contact, the spots can slip or stick, dependingwhether the instantaneous friction bound is exceeded or not. These many options aretaken into account numerically using a case distinction of the equations of motion asshown in section 2.3.3. A detailed description of the procedures in pseudo-code-notationis also given in appendix B.4.

ϕ

x

zxs

κ4released

fN2

fT 2

1

2

Fig. 4.4: free body diagram of the non-dimensional system, where xs denotes the displace-ment of the base. Due to the rotation, the left contact is released from the substrate

Assuming small rotations, i.e. ϕ ≪ 1, and the springs to be undeflected for x, z, ϕ = 0,Newton’s second law [95] yields the equations of motion (EoM) for the system depictedin Fig. 4.4 as:

x′s = κ6 , (4.1.16)

x′′ = −ux1 − ux2 + κ1 (xs − x + κ8ϕ) , (4.1.17)

z′′ = −κ2uz1 − κ2uz2 + κ4 , (4.1.18)

ϕ′′ =1

κ7

(

−κ2κ3uz1 (1 − ϕ) + κ2κ3uz2 (1 + ϕ) − ux1

(

κ23 + ϕ

)

− ux2

(

κ23 − ϕ

))

− 1

κ7

(

κ1

(

κ23κ8 + ϕ

)

(xs − x + κ8ϕ))

. (4.1.19)

Equation (4.1.16) is the result of introducing a new variable xs = κ6∆t, what removesthe time dependency and enhances the computation [96]. The motion of the system iscomputed using a stepwise numerical integration scheme. The basic principle may be

67


explained with reference to the simple Euler method [97]. According to this, for a givenderivative y = f (y, t) and the initial value y (t) the solution in the next step y (t + ∆t)can be approximated as:

y (t + ∆t) = y (t) + ∆tf (y, t) , (4.1.20)

where ∆t denotes the time step. The so called Velocity-Verlet algorithm is used todetermine the motion of the system from equations (4.1.16) - (4.1.19), where a detaileddescription of the scheme is given in Appendix B.3. The algorithm is widely used inmolecular dynamics (MD) to compute the motion of molecules on basis of Newton’ssecond law. In MD the forces acting on a molecule in time t + ∆t are given by apotential, i.e. they are a function of the position of the molecule at time t + ∆t. Inthe same way the state of motion of the system x, z and ϕ in the time step t + ∆t iscomputed firstly. The normal and tangential spring deflections uz1/2 and ux1/2 at timet + ∆t, i.e. the contact forces, are computed afterwards as described in Appendix B.4.These determine the velocities at time t + ∆t that in turn determine the state of motionin time-step t + 2∆t and so on.

0 2 4 6 8 10−2

0

2

time t∗ = t − t0

dynam

icquan

titi

es

∆xzϕ

fs

Fig. 4.5: steady state motion for pa-rameter set A. All quantities are normal-ized as ∆x = (x − xs) /µκ4, z = z/zstat,ϕ = ϕ/µκ3κ4 and fs = fs/µκ4

0 2 4 6 8 10

−2

0

2

time t∗ = t − t0

dynam

icquan

titi

es

∆xzϕ

fs

Fig. 4.6: steady state motion for pa-rameter set B. All quantities are normal-ized as ∆x = (x − xs) /µκ4, z = z/zstat,ϕ = ϕ/µκ3κ4 and fs = fs/µκ4

Using the numerical integration procedures described in appendix B.3, the motion ofthe system in steady state, i.e. x (t) , z (t) and ϕ (t), as well as the spring force fs (t) aredetermined for various sets of parameters. The initial condition are as follows:

x (0) = 0 , x′ (0) = 0 ,z (0) = zstat , z′ (0) = 0 ,ϕ (0) = 0 , ϕ′ (0) = 0 ,xs (0) = 0 ,

(4.1.21)

where zstat denotes the static displacement yielded with equations (4.1.18) and (B.4.1)

68


from appendix B.4:

z′′ = 0 ⇒ zstat =κ4

2κ2. (4.1.22)

In order to determine the frictional resistance the spring force fs is computed as:

fs (t) = κ1 (xs (t) − x (t) + κ8ϕ (t)) . (4.1.23)

This excitation force corresponds to the macroscopic resistance force of the overall sys-tem. Finally, with 〈.〉 being the time average, the effective coefficient of friction iscomputed as:

µe =〈fs (t)〉

µκ4. (4.1.24)

Figures 4.5 and 4.6 depict the normalized steady state motion for two parameter sets asa function of time t∗ = t − t0. Here t0 indicates a time at which the transient process isalready completed. The two sets of parameters are as follows:

set A: κ1−8 =(

2, 1.2, 0.5, 10−4, 0.5, 10−4, 0.42, 0.5)

, (4.1.25)

set B: κ1−8 =(

2, 1.2, 0.5, 10−4, 0.68, 10−4, 0.42, 1.18)

. (4.1.26)

Despite the fact that the parameter sets only differ slightly, i.e. µ = 0.5 and κ8 = 0.5for set A versus µ = 0.68 and κ8 = 1.18 for set B, the steady state motion of the systemdiffers strongly. This becomes clearer considering the vertical displacement z that isrepresented by the red lines in Fig. 4.5 and Fig. 4.6. For set A applies z = const.,whereas for set B z follows an harmonic oscillation. A more detailed analysis of theinfluence of the different parameters is given in the following section 4.2.

4.2 System Dynamics and Frictional Resistance

Using the simulation model as described in section 4.1 appropriate parameter ranges ofκ1−8 are identified that lead to a reduction of the effective frictional resistance µe. Thisis followed by a more detailed analysis in order to identify the parameter combinationnecessary for a (theoretical) maximal reduction. The experimental setting that is de-scribed in section 5.2 is designed such that it matches this parameter combination asclosely as possible.

4.2.1 Influence of the Parameters

The non-dimensional model is described by eight parameters, each of it potentially influ-encing the motion and in turn the effective frictional resistance of the system. Startingpoint is the determination of appropriate basic points of the parameter space. These aredescribed in the following.

69


The ratio of the base-spring stiffness and the contact stiffness κ1 can be interpretedas a measure for the stiffness of the system. Hence, high values correspond to a stifftribological system such as a disc brake. One example for a soft system is a pin-on disctribometer with flexible arm. The range of κ1 is assumed to be:

κ1 =ks

kx⇒ κ1 ∈

[

10−1, 10]

. (4.2.1)

The ratio κ2 depends on the contact geometry. A narrow bandwidth is considered:

κ2 =kz

kx⇒ κ2 ∈ [1, 1.4] . (4.2.2)

Parameter κ3 denotes the ratio of height and width of the rigid body, i.e. the slenderness.Medium values are considered:

κ3 =a

b⇒ κ3 ∈ [0.5, 1.5] . (4.2.3)

Parameter κ4 lacks an illustrative physical meaning. However, one can estimate it as-suming a specimen of mass m = 100 g and half-width b = 1 cm and a tangential contactstiffness of kx ∈

[

104, 105]

N/m (see section 5.2.1):

κ4 =mg

kxb⇒ κ4 ∈

[

10−4, 10−2]

. (4.2.4)

In this first step, medium values of the microscopic coefficient of friction are assumed:

µ ∈ [0.2, 0.6] . (4.2.5)

The parameter κ6 depends inter alia on the speed of the constantly moving base v0.Assuming a velocity range of v0 ∈ [1, 100] mm/s it results to:

κ6 =v0

b

√

m

kx⇒ κ6 ∈

[

10−4, 10−2]

. (4.2.6)

Assuming a rectangular shape of the rigid body and taking into account2 (4.2.3), theratio of the rotational inertia results to:

κ7 =Θ

mb2=

4

12

(

1 + κ23

)

⇒ κ7 ∈ [0.5, 1.5] . (4.2.7)

Finally, κ8 denotes the ratio of the lever arm and the height. Allowing the spring to beapplied underneath the centre of gravity, the range of κ8 results to:

κ8 =h

a⇒ κ8 ∈ [−0.8, 0.8] . (4.2.8)

2It is important to emphasize that κ7 is an independent parameter and is not influenced by κ3, asthe exact shape of the rigid body might differ from the rectangular shape.

70


Following a straightforward approach one would for instance analyse ten different valuesfor each parameter. This would result in 108 different parameter combinations. Con-sequently, given the computation time for one combination of approximately 90 s, thiswould finally require an overall computation time of 300 years if one uses a standardPC and simulate one combination after the other. To avoid this, the so called design ofexperiments (DoE) approach is used, which is introduced in detail at appendix A.2. Thismethod systematically minimizes the effort needed for a sufficiently accurate analysis.A so-called full-factorial experimental schedule is used that considers all possible basicpoints in parameter space. Thus, three values of each parameter (minimum, intermedi-ate, maximum) are taken and the motion of the system in steady state is computed togive the effective coefficient of friction. Overall, this leads to 38 = 6561 different combi-nations. In order to enhance the overall computation time, the calculation is performedin parallel on a graphics processing unit (GPU). Using a Geforce GTX 560 GPU and astep size of ∆t = 1 ·10−3 the whole calculation takes about 15 min for a non-dimensionalperiod of observation of T = 4000. Finally, the main effect is calculated, which is themean of all 6561/3 = 2187 combinations in which one specific parameter is held con-stant. The so called interaction effect, which describes the interaction between differentparameters, is not considered here, since the high number of combinations preclude ameaningful evaluation. For computation of the main effect only solutions without tiltingof the specimen are considered, see also section 4.2.2. For this purpose, solutions withhigh rotational amplitudes are excluded from the evaluation. Figure 4.7 depicts the maineffect on the effective coefficient of friction µe. The overall influence of factors is ratherlow, as the maximal reduction is in the range of 10 %.

1 1.2 1.4κ2

0.1 1 100.88

0.9

0.92

0.94

0.96

0.98

κ1

µe

0.5 1 1.5κ3

10−4 10−3 10−2

κ4

0.2 0.4 0.60.88

0.9

0.92

0.94

0.96

0.98

µ

µe

10−4 10−3 10−2

κ6

0.5 1 1.5κ7

-0.8 0 0.8κ8

Fig. 4.7: main effect-plot of the eight parameters of influence κ1−8 on the effective coeffi-cient of friction µe

71


On the one hand, this may mean that the individual influence of all parameters is low andthere do not exist any points in the parameter space that lead to a strong reduction. Onthe other hand, this may indicate that only a few parameters are from great influence,whereas the majority has almost no influence. Thus, a strong reduction would onlyoccur in a few points in parameter space. In this work the DoE analysis rather servesas a tool for the identification of convenient parameter combinations. Despite the factthat a further analysis is given later in section 4.2.2 and section 4.2.3, the results arebriefly interpreted. However, it must emphasized that the results of the DoE should beconsidered with caution, as the overall effect is rather low.More specifically, it turns out that µe is low for κ1 around 1, meaning that a similarityof the excitation and contact stiffness is convenient. This resembles to the model ofMartins et al. in which a large stiffness parameter sM or a small sM combined with alarge damping lead to an apparently smooth sliding with an effective coefficient of frictionlower than the static one [10]. The ratio κ2 has a low influence. In contrast, κ3 is fromgreat influence indicating that a more compact shape of the rigid body decreases µe. Ahigher κ3 leads to a higher rotational moment of the spring force fs in comparison tothe contact forces. Conversely, the friction increases with increasing κ4. Thus, low ratioof mass and contact stiffness reduces the effective friction in this system. The influenceof the microscopic friction µ is relatively high, where µe decreases with increasing µ.This effect is consistent with the beam model by Adams [47] and can be explained byan increasing interplay of the tangential forces and the rotational moment that leadsto self-excited oscillations. The friction also decreases for increasing κ6 indicating thatthe effective friction decreases with increasing velocity. This effect also occurs in theslip wave model of Adams [50] and in the rigid body model of Martins et al. [10]. Inaddition, the effect of decreasing friction with increasing velocity was experienced innumerous experiments [77, 78, 79] where an overview is given for example in [9]. Theinfluence of κ7, which depends on the rotational inertia, is relatively poor. In contrast,the lever arm of the base, which is represented by κ8, is from great influence. Theeffective friction decreases with increasing lever arm, thus with an increasing momentof the spring force fs with respect to the contact spots. This increases the couplingbetween the rotational moment and the friction forces and is consistent with the beammodel [47].In addition, the main effect on the minimal vertical amplitude in steady state zmin/zstat iscomputed as given in Fig. 4.8. Here zstat is the static displacement as defined in (4.1.22).It shows that the amplitudes are negative and reach values of -100. A negative zmin

indicates jumping of the rigid body, i.e. a total release of the contact spots. In addition,a high magnitude of zmin indicates instabilities and self-excited oscillations of the rigidbody in the normal direction that are caused by the interplay of the friction force and therotational moment. However, it should again be emphasized that the high magnitudescan indicate two scenarios. Namely, a small number of parameters combinations canexhibit very high corresponding magnitudes while the remaining combinations exhibitno jumping at all. Or a large number of parameter combinations can exhibit similarlyhigh amplitudes. In both cases, jumping must not necessarily occur in all combinations.

72


1 1.2 1.4κ2

0.1 1 10−100

−80

−60

−40

−20

κ1

z min

/z s

tat

0.5 1 1.5κ3

10−4 10−3 10−2

κ4

0.2 0.4 0.6−100

−80

−60

−40

−20

µ

z min

/z s

tat

10−4 10−3 10−2

κ6

0.5 1 1.5κ7

-0.8 0 0.8κ8

Fig. 4.8: main effect-plot of the eight parameters of influence κ1−8 on the vertical minimalamplitude zmin/zstat

In detail, it shows that κ2, κ3 and κ7 all have a weak effect on the vertical amplitude.The absolute value increases with decreasing κ1, what shows that the vertical motionis stronger, the weaker the guidance of the spring. Decreasing κ4 strongly increases themagnitude of zmin. Thus, a heavier specimen jumps less strongly for the same contactstiffness and width b. The magnitudes of zmin strongly increase with µ. This may beexplained with the effect that a higher friction increases the coupling between lateraland vertical translation and rotation, what leads to higher displacements in the verticaldirection. This is again consistent with the Adams beam model where increasing µ leadsto increasing instabilities [47]. The same effect occurs in the Martins et al. model, wherethe normal displacement increases with the friction parameter fM [10]. Parameter κ6

has a strong effect, indicating that a higher velocity generates higher vertical jumps.One can assume that a higher velocity simply increases the kinetic energy of the systemthat is then transferred to the vertical motion through the coupling effect. In addition,it shows that the vertical amplitudes are weak for κ8 = 0.

Taken together, the DoE analysis shows that a relatively narrow parameter range needsto be matched in order to induce a reduction. Especially for the parameters κ4, µand κ6 the reduction is accompanied with an increasing amplitude of the oscillation inthe vertical direction. Again it should be emphasized that the DoE analysis is usedto determine parameter combinations that enable a significant reduction. It is not theintention to give a detailed analysis at this point. However, the results are consistentwith those found in the literature.

73


4.2.2 Reduction of the Frictional Resistance

In the following some of the parameters are kept constant for a more detailed analysis.The results of section 4.2.1 show that a significant reduction requires κ1 (macroscopicstiffness) around 1 and low values of κ4 (normal force vs. contact stiffness times halfwidth). In addition, the influence of κ2 and κ7 is relatively poor and high values ofκ6 lead to a strong amplification of the vertical amplitudes. These and the parameterκ3 are kept fixed, where the exact values are listed in Tab. 4.2.1. A low value for κ6

is chosen in order to induce only small vertical amplitudes in the range of the staticdisplacement zstat. This maintains the microscopic character of the walking effect. Theremaining parameters are the microscopic friction µ and the position of the lever arm κ8.According to the DoE analysis, both have a significant influence on the efficient friction.In addition κ8 is relatively easy to vary in real experiments and the influence of µ wasalready examined by other authors [10, 47] what gives the possibility for comparison.

Tab. 4.2.1: parameters that are constant in the detailed parameter study

parameter κ1 κ2 κ3 κ4 κ6 κ7

value 2 1.2 0.5 10−4 10−4 0.42

A parameter study for 1600 combinations of the two remaining factors is conducted.Again, the influence on the effective friction µe is examined, which is shown in Fig. 4.9.In addition, the influence on the minimal vertical amplitude of the motion of the centreof gravity of the specimen is examined, which is shown in Fig. 4.10. In both figuresoccur three characteristic parameter ranges, namely the no effect-range, the reductionrange and the critical range. These can be explained as follows.

0 0.5 1 1.5 2

0.5

1

1.5

2

critical

reduction

no effect

µc

κ8

µ

0.2

0.4

0.6

0.8

1

µe

Fig. 4.9: parameter study of κ8 and µfor the effective coefficient µe. The stardenotes the maximal reduction of 98 %.The dash-dot line gives µc of (4.2.13)

0 0.5 1 1.5 2

0.5

1

1.5

2

µc

critical

zmin < 0

zmin = zstat

κ8

µ

−30

−20

−10

0

zmin

zstat

Fig. 4.10: parameter study of κ8 andµ for the minimal amplitude zmin/zstat.In the reduction range applies zmin < 0.The dash-dot line gives µc of (4.2.13)

74


No Effect Range Here no reduction of the macroscopic frictional resistance occursat all, i.e. µe = 1. This range is indicated by the solid line in Fig. 4.9. The comparisonof Fig. 4.9 and Fig. 4.10 shows that this parameter range coincides with a region oflow vertical amplitudes or even of no vertical vibrations at all. The region of constantvertical displacement is delimited by the solid line in Fig. 4.10. This indicates thatvertical vibrations play an important role in the reduction effect. Without verticalvibration, no reduction occurs at all. This is consistent with the known experiments ofTolstoi et al. [39, 40, 93].

Critical Range The reason of the critical behaviour of the system is caused by atilting of the rigid body over spot 2. It is indicated by the white regions in Fig. 4.9 andFig. 4.10. The simulation is stopped in this case, because the model as stated in (4.1.16)- (4.1.19) is no longer valid as the magnitudes of ϕ no longer permit linearization of thetrigonometric functions. In order to give an explanation for this effect the dimensionfulsystem as depicted in Fig. 4.2 (b) is used. Consider a tilted position of the rigid bodysuch that contact 1 is completely released. This is situation is sketched in Fig. 4.11where a free body diagram is given. The only forces acting on the rigid body are thespring force Fs, the force of gravity mg and the forces in the right contact 2.

Φ

≈ a + h≈ b

Fs

mg

FN2

FT 2

1

2released

Fig. 4.11: free body diagram of the system in a tilted position

Using the principle of angular momentum with respect to contact 2 and assuming smallrotations, i.e. Φ ≪ 1, gives:

Θ(2)Φ = −Fs (h + a) + mgb . (4.2.9)

A further tilting in the clockwise direction requires a negative angular acceleration Φ < 0,what yields:

Fs (h + a) > mgb . (4.2.10)

Assuming sliding of spot 2 and a constant velocity v0, the spring force Fs must matchthe tangential force FT 2. In turn, FT 2 corresponds to the actual friction bound, i.e.FT 2 = µFN2. Given the free body diagram of Fig. 4.11 and the assumption Z = 0 thespring force in the tilting state results to:

Fs = FT 2 = µFN2 = µmg . (4.2.11)

75


Inserting this into (4.2.10) gives the condition for the coefficient of friction for which therigid body tilts over spot 2:

µ >mgb

mg (h + a)=

b

(h + a). (4.2.12)

Finally, using the definition of the non-dimensional parameters as listed in Tab. 4.1.1,the non-dimensional form of the critical coefficient µc for the limiting case yields:

µc =1

κ3 (1 + κ8). (4.2.13)

Relation (4.2.13) is denoted by the dash-dot line in Fig. 4.9 and Fig. 4.10 and is ingood agreement with the simulation results. Within the critical limit, the rotationalamplitudes where small, i.e. ϕ < 1 · 10−3. A similar type of critical system behaviourwas also studied by Martins et al. for their sliding block model. In this, the maximumvalue fM of the friction parameter fM was chosen such that steady sliding equilibriumceases to be possible due to tumbling of the block [10]. The case κ3 = 1 and κ8 = 1exactly corresponds to the case fM = 1/hM , where hM is the ratio of height and lengthof the block. They conclude that no one would run a comparable experiment allowingfor the occurrence of such large oscillations. However, they point out that the same maynot be true for instance in a pin-on-disk tribometer having very flexible arms and a smallcontact region.

Reduction Range For a specific range of parameters occurs a significant reductionof µe over 50 %. The comparison of Fig. 4.9 and Fig. 4.10 shows that the reduction co-incides with negative vertical amplitude zmin. This indicates that a total release of thecontact spots, i.e. jumping, is an important prerequisite for a reduction in this system.Thus, one reason for the reduction are oscillations of the rigid body in the vertical di-rection. This is consistent with the well-known experiments of Tolstoi et al. [39, 40, 93]and with theoretical works [10, 41, 42, 47]. However, the amplitudes are of the orderof the static vertical displacement. Hence, in a real system these vibrations are from amicroscopic character and can be superposed unnoticed to an apparently smooth slidingmotion. The maximal reduction of 98% is denoted by the star and occurs at κ8 = 1.18and µ = 0.68. The corresponding minimum amplitude is zmin/zstat = −0.42.

For a further analysis, the relative stick time of the contact spots is computed, which isthe ratio of the time a specific contact sticks Ts1/2 and the period of observation T :

ts1/2 =Ts1/2

T. (4.2.14)

The stick times are depicted in Fig. 4.12 and Fig. 4.13.

76


0 0.5 1 1.5 2

0.5

1

1.5

2

critical

reduction

no effect

µc

κ8

µ

0

0.05

0.1

0.15

0.2

ts1

Fig. 4.12: parameter study of κ8 and µfor the relative stick time ts1 of contact 1.In the reduction range applies ts1 ≈ 15%

0 0.5 1 1.5 2

0.5

1

1.5

2

critical

reduction

no effect

µc

κ8

µ

0

0.05

0.1

0.15

0.2

ts2

Fig. 4.13: parameter study of κ8 and µfor the relative stick time ts2 of contact 2.In the reduction range applies ts2 ≈ 20%

It shows that sticking coincides with the reduction with ts1 ≈ 15% and ts2 ≈ 20% in thereduction range. In contrast, no sticking occurs in the no effect range. Thus, stickingis important for the reduction. There are three explanations for this. Firstly, stickingleads to storage of elastic energy in the contacts. This energy is needed for the jumpingthat occurs in the reduction range, as shown in Fig. 4.10. Secondly, the concept ofmicro-walking implies that only one spot is sticking, while the other one is slipping or iscompletely released from the substrate as introduced in section 4.1. And thirdly, stickingmay decrease the average value of the friction force as observed by Martins et al. [10].Furthermore, the Pearson correlation coefficients r1 and r2 are computed to examinethe linear correlation between the velocity of a spot s′

1/2 and the corresponding normalspring deflection uz1/2, i.e. the corresponding normal force. More information on thecorrelation coefficients is given in appendix A.1.4. The velocities are computed as thecentral difference of the motion of the spots divided by the time step ∆t:

s′1/2 (t) =

1

2∆t

(

s1/2 (t + ∆t) − s1/2 (t − ∆t))

, (4.2.15)

where the motion of spots s1/2 is given as:

s1/2 (t) = x (t) + ϕ (t) − ux1/2 (t) . (4.2.16)

Figures 4.14 and 4.15 give the correlations r1 and r2 for the spots. The case r = 1describes a positive linear correlation between two quantities. A comparison of thedifferent parameter ranges in both figures shows that the correlations are about -0.5 inthe reduction range. This indicates a negative linear correlation of spot velocities andnormal forces as low forces coincide with high velocities and vice versa. This means thatthe spots make most of their motion while the corresponding normal forces are low. Inaddition, they stick, i.e. s′

1/2 = 0, while the corresponding normal force is high. Firstly,this indicates that the specimen walks as proposed in section 4.1. Secondly, this indicatesthat the rigid body and its contact spots make most of the forward motion while being

77


totally released from the substrate. This is in particular confirmed by comparison ofFig. 4.9 and Fig. 4.10, what shows that the reduction coincides with negative verticalamplitudes zmin, i.e. jumping of the rigid body.

0 0.5 1 1.5 2

0.5

1

1.5

2

critical

reduction

no effect

µc

κ8

µ

−0.8

−0.6

−0.4

−0.2

0

r1

Fig. 4.14: parameter study of κ8 and µfor the correlation coefficient r1 of veloc-ity s′

1and normal deflection uz1 of con-

tact 1. In the reduction range appliesr1 . −0.5

0 0.5 1 1.5 2

0.5

1

1.5

2

critical

reduction

no effect

µc

κ8

µ

−0.8

−0.6

−0.4

−0.2

0

r2

Fig. 4.15: parameter study of κ8 and µfor the correlation coefficient r2 of veloc-ity s′

2and normal deflection uz2 of con-

tact 2. In the reduction range appliesr2 . −0.5

4.2.3 Vanishing Frictional Resistance

In order to illustrate the proposed effects, the maximal reduction case is considered,which is symbolized by the black star in the figures 4.9, 4.10 and 4.12-4.15. The corre-sponding parameter combination is given as:

κ1−8 =(

2, 1.2, 0.5, 10−4, 0.68, 10−4, 0.42, 1.18)

. (4.2.17)

The maximum is 98% and the corresponding minimal amplitude is zmin/zstat = −0.42.This indicates jumping of the rigid body, i.e. a complete release of the contact spots.In order to examine the influence of the initial conditions on the steady state and thetransient process, the DoE approach as described in appendix A.2 is used. Three initialvalues for each motion variable are used what gives 37 = 2187 combinations of initial con-ditions. The initial displacements are chosen on basis of the vertical static displacementas defined in equation (4.1.22):

x (0) , z (0) , ϕ (0) , xs (0) ∈ [−zstat, 0, zstat] , (4.2.18)

and the base velocity, i.e. the parameter κ6, gives the initial velocities:

x′ (0) , z′ (0) , ϕ′ (0) ∈ [−κ6, 0, κ6] . (4.2.19)

The non-dimensional mean kinetic energy of the steady state serves as a measure:

ekin =1

κ4zstat

1

2

(

x′2 + z′2 +κ7

κ23

ϕ′2)

. (4.2.20)

78


For comparison, the non-dimensional initial mechanical energy is computed:

em (0) =1

κ4zstat

1

2

(

u2x1 + u2

x2 + κ2

(

u2z1 + u2

z2

)

+ κ1 (xs − x + κ8)2 − κ4z)

+ ekin (0) . (4.2.21)

Figure 4.16 gives the main effects of the initial conditions on the steady state kineticenergy, which is depicted by the red line, and on the initial mechanical energy, whichis represented by the blue line. It shows that the steady state kinetic energy is notinfluenced by the initial conditions. One can argue that the frictional damping leadsto a decay of the oscillations that are caused by the initial mechanical energy. For theconsidered range of initial conditions the system reaches practically the same steadystate.

−zstat 0 zstat

z (0)

−zstat 0 zstat3

4

5

6

7

x (0)

e kin

,em

(0)

−zstat 0 zstat

ϕ (0)

−zstat 0 zstat

xs (0)

−κ6 0 κ6

z (0)

−κ6 0 κ63

4

5

6

7

x (0)

e kin

,em

(0)

−κ6 0 κ6

ϕ (0)

ekin

em(0)

Fig. 4.16: main effect-plot of the initial conditions on the mean kinetic energy in steadystate (red line) and the corresponding initial mechanical energy (blue line)

Figure 4.17 depicts the phase-space diagram for the relative displacement x − xs for themaximal reduction combination. The black dot depicts the initial conditions x (0) = 0and x′ (0) = 0. After the blue coloured transient process, the system reaches a stablelimit cycle (lc), which is dark coloured. Several dynamic quantities as a function of thetime t∗ = t−t0 in steady state are displayed in Fig. 4.18. Here t0 indicates a time at whichthe transient process is already completed. The difference ∆x = x (t) − x (t0) is shown,to give all quantities in one plot. It shows that the centre of gravity exhibits a harmonicoscillation around the static displacement zstat, what leads to highly varying normaldeflections of the springs. As one expects, their maximum approximately coincides with

79


the maximum vertical displacement. The small shift between uz1 and uz2 is caused bythe rotation ϕ.

−4 −2 0

0

2

(x − xs)/µκ4

x′ /

κ6

lc

Fig. 4.17: phase space diagram of thelateral motion for the maximal reductionrange. The system reaches a stable limitcycle (lc)

0 1 2 3 4

0

2

4

(1)

(2)

(3)

(4)

t∗ = t − t0

dynam

icquan

titi

es ∆xzϕ

uz1

uz2

Fig. 4.18: motion of the system forthe maximal reduction range. Dy-namic quantities are normalized as ∆x =∆x/µκ4, z = z/zstat, ϕ = ϕ/µκ3κ4 anduz1/2 = uz1/2/µκ4

zzstat

(1)(1)

(2) (3)

(4)

ff

Fig. 4.19: phases of motion of the specimen in the vanishing reduction range. The rigidbody follows harmonic oscillation in the vertical direction. The rotation is not shown asϕ < 1 · 10−3 applies in steady state

The vertical motion is highly synchronized with the lateral translation x. More specif-ically, the rigid body makes most of the lateral displacement when the vertical dis-placement is negative, i.e. the body is jumping, and the contacts are released from thesubstrate. One can identify four specific states as marked in Fig. 4.18 and Fig. 4.19:

(1) start of sticking phase: sticking contact spots, downward movement and increasingnormal forces, only minor lateral motion

(2) turning point: end of downward movement, sticking contact spots, high normalforces, storage of elastic energy

(3) end of sticking phase: upward movement and start of forward movement

(4) microscopic jump: release of the spots and zero contact forces, fast forward (ff)motion of the rigid body

80


The motion can be characterized as a highly synchronized vertical and lateral vibrationwith alternating jumping and sticking phases of the rigid body. It is this particularsystem behaviour that leads to an almost vanishing frictional resistance. This matchesthe effect of micro-walking as introduced in section 4.1.

4.2.4 Summary of the Theoretical Results

Considering the overall results of section 4.2, one can finally identify three main factorsthat are responsible for the reduction effect of the frictional resistance:

1. The coupling of the tangential, vertical and rotational degrees of freedom causesself-excited oscillations in the vertical direction. As the friction forces in the con-tact spots are mainly responsible for this, the amplitudes grow with increasingmicroscopic friction and lever arm of the base spring with respect to the contactspots. This effect is consistent with theoretical works of Martins et al. [10] andAdams [47].

2. The experimental works of Tolstoi et al. [39, 40, 93] showed that the normal vibra-tions themselves already cause a reduction due to the non-linearity between normalseparation and normal force. However, in the present work a linear dependencybetween spring deflections and contact forces is assumed. Hence, the reduction iscaused by another effect.The most significant effect in the micro-walking machine is the reduction of theoverall resistance force that is induced by the strong correlation between low oreven zero contact forces and forward motion respectively high normal forces andsticking contacts. This corresponds to the micro-walking as proposed in section 4.1.

3. Jumping and fast forward motion of the rigid body requires energy that is storedin the elastic springs. This is maintained by a characteristic alternation betweenstorage and motion phase: the spots stick and energy is conserved that is thanused for the vertical jumps and the tangential fast forward motion. Furthermore,sticking simply decreases the average value of the friction force as observed byMartins et al. [10].

One can conclude that the dynamic mode that is responsible for the reduction is theproposed micro-walking effect, i.e. a convenient synchronization of lateral and verticalmotion and a jumping of the rigid body with released contacts.

81


4.3 Experiments

The results of section 4.2 give clear but also strict guidelines for the design of a realtechnical system with the effect of frictional resistance reduction through self-excitedoscillations. However, there exist certain experimental limits, as discussed in detail insection 5.2. Firstly, there occur differences in the geometry and the contact configurationbetween theoretical and real system. These are mainly caused by the fact that the two-dimensionality of the theoretical model cannot be simply transferred to the real world inparticular due to alignment problems. Secondly, there exist several practical limitationswhich make it impossible to reach the values of the non-dimensional parameters thatare required for the almost vanishing friction. For instance, simultaneously fulfillingthe conditions κ1 ≈ 1 (ratio of spring stiffness and contact stiffness) and κ4 ≈ 1 · 10−4

(normal force vs. contact stiffness times half width) is very hard in practice as discussedin section 5.2.3. Thirdly, the results show that there exists a critical coefficient of frictionµc that depends on the geometry respectively the point of application of the spring forceas shown in section 4.2.2. To avoid the critical range, the experiments are limited tonegative values of κ8 (ratio of lever arm of the base), i.e. the point of application offs is always underneath the centre of gravity. According to equation (4.2.13) this willenhance the stability of the experiment with respect to the rotation, i.e. will avoidtumbling of the rigid body, but will also decrease the reduction. Finally, some of theparameters are easier to vary in the experiments than others. For instance, it does notrequire much effort to change κ8 or κ6 (velocity of the base v0). In contrast, changing κ3

(slenderness of the rigid body) or κ7 (ratio of rotational inertia) requires an update ofthe geometry and changing µ requires a different combination of materials. This wouldlead to undesired changes in several other parameters as κ1, κ4 and κ6 as they dependon material properties. Overall, five parameter sets are considered which are named set1–5. The physical parameters are given in section 5.2 in Tab. 5.2.1.

4.3.1 Experimental Results

Due to the reasons stated above, i.e. the effort for a variation of certain parameters,firstly significantly different parameter ranges without intermediate combinations arecompared as listed in Tab. 4.3.1. In order to identify convenient parameter combinations,the extended model as introduced in section 5.2.3 and appendix B.4 is used. It showsthat within the experimental limits, the theoretical maximal reduction is about 50%.Thus, the reduction in the experiment is expected to be lower than in the theoreticalmodel of section 4.1. However, the contact stiffness was initially underestimated. Onthis basis an insufficient spring stiffness was chosen such that the stiffness parameterwas κ1 = 0.1 instead of the intended κ1 = 2. Set 1 meets the theoretical maximal rangewithin the experimental limits except for κ1. Set 2 has unsuitable geometry and inertiaproperties, i.e. κ3 is inconveniently changed. Set 3 and set 4 correspond to the previousones with low κ1, i.e. a very soft base-spring. For this purpose, a spiral spring withstiffness ks−SP = 60 N/m is used. In the experiments, the base velocity v0 is varied,which is linear proportional to the non-dimensional parameter κ6.

82


Tab. 4.3.1: parameters of the different sets used in the experiment. Parameters κ2 and κ8

are the same for all

set κ1 κ2 κ3 κ4 µ κ7 κ8

1 ≈ 0.1 1.33 1.51 2 · 10−4 0.79 0.95 -0.47

2 ≈ 0.1 1.33 0.85 1 · 10−4 0.79 0.25 -0.47

3 ≈ 10−4 1.33 1.51 2 · 10−4 0.68 0.95 -0.47

4 ≈ 10−4 1.33 0.85 1 · 10−4 0.68 0.25 -0.47

The spring force Fs (t) is measured and the effective coefficient of friction is computedas defined in equation (5.2.2). Figure 4.20 depicts the results of the different sets. Forthe sake of clarity, all curves are plotted over v0 and not over κ6 as this parameter iscross-influenced by other quantities that change between the sets. The reduction forset 1 is about 54%, meaning that the effective friction µe is more than halved even forκ1 = 0.1. In contrast, no significant reduction applies for the other sets. The effectivefriction slightly varies within a range of approximately 10% of µ. One can conclude,that the effect of friction minimization as introduced in section 4.2 does not occur forsets 2-4, thus µe ≈ 1. This means, just as with the theoretical results, that in orderto induce the friction minimizing effect, a very narrow parameter range has to be met.In addition, the effect hardly depends on velocity but only occurs if a certain velocitylimit is exceeded as can be seen for set 1 where µe is halved only for v0 ≥ 15 mm/s.However, the experimental results show that it is possible to induce the effects introducedin section 4.2 in a real system.

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

base velocity v0 [mm/s]

effec

tive

fric

tion

µe

set 1set 2set 3set 4

Fig. 4.20: effective coefficient of friction µe for varying base velocity v0 and differentparameter sets

In the experiments, the only time varying value that is measured is the spring forcein the base. The normal oscillations have not been measured. Nevertheless, one can

83


assume that whenever instabilities occur in the normal direction, i.e. strong oscillations,they will also affect the tangential direction due to the coupling effect. Thus, self-excitedoscillations in the normal direction must be detectable in the time response of the basespring. Figure 4.21 displays the time response of the spring force Fs for set 1 and set2. It shows that in case of set 1, which is represented by the blue line, the reductionis accompanied with self-induced oscillations as Fs oscillates with high frequency andmagnitude. The force reaches negative values, meaning that the specimen pushes thebase point of the spring. For set 2, which is represented by the red line, the forceis not oscillating and is positive all the time. In this case the instability, i.e. self-excited oscillations in the vertical direction with high amplitude, do not occur at all.Consequently no reduction is induced.

8.2 8.3 8.4 8.5−1

0

1

2

time t[s]

spri

ng

forc

eF

s/√

2µm

g

set 1set 2

Fig. 4.21: time course of the springforce for set 1 and set 2 for v0 = 20 mm/s.Instability occurs only for set 1 but stopstemporarily

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

base velocity v0 [mm s−1]

tim

eof

inst

abilit

yt i

s/T

Fig. 4.22: relative time of instabilitytis as a function of the base velocityv0 for experimental set 1. Star showsv0 = 20 mm/s

Additionally, it shows that the vibrations of the force of set 1 significantly decreasetemporarily. This can be seen in Fig. 4.21 for t > 8.4 s. Consequently, the instability aswell as the reduction effect temporarily pauses. In contrast, such an effect does not occurin the simulation at all. One can conclude that the pausing is caused by experimentalimperfections rather than self-stabilization. The geometry and the microscopic friction,which is influenced inter alia by the roughness and contamination by dust, will slightlychange when the substrate is moved along the specimen. In order to determine the ratioof the time the instability occurs, i.e. a significant increase of the amplitudes of Fs withnegative values, and the time of observation tis/T , the revolving variance var (Fs (t)) ofthe friction force is considered. Within the Dirichlet window with size 2tD of the timeresponse, the variance is defined as:

var (Fs (t)) =

√

√

√

√

1

2tD/∆t

ti=t−tD∑

ti=t−tD

Fs (ti) − Fs (t) , (4.3.1)

84


where Fs (t) denotes the corresponding mean within the window:

Fs (t) =1

2tD/∆t

ti=t−tD∑

ti=t−tD

Fs (ti) . (4.3.2)

Here Fs (ti) denotes the spring force of the measurement sample i at time ti. The window2tD corresponds to approximately 10 characteristic periods of Fs (t). As a criterion,only those time sections are regarded as instable in which var (Fs (t)) > 0.1 applies.In addition, for determination of the effective friction µe only instable time sectionsare taken into account. Figure 4.22 gives the relative time of instability tis/T for set1. It shows that tis ≈ T applies if a certain velocity limit is exceeded. Additionallythe self-induced oscillations highly coincide with minimized friction as can be seen bycomparison with Fig. 4.20. The instability is thus crucial for the reduction effect. Thisconfirms the results of the theoretical model given in section 4.2.

4.3.2 Comparison of Experiment and Model

The results of section 4.2.2 indicate that the application point of the spring is fromgreat influence for the reduction. For this reason, the joint influence of the velocity v0,i.e. parameter κ6, and the position of the lever arm, which refers to κ8, is examined.Four values of κ8 are considered. In the experiment this is simply realized by severaldifferently positioned mounting holes, as shown in Fig. 4.23 (a). The specimen is thenrotated such that the line between the selected mounting hole and the centre of gravity isvertical. As sketched in Fig. 4.23 (b), the experiments are restricted to values of κ8 < 0in order to reduce unwanted instabilities of the motion, i.e. tumbling and tilting of therigid body.

centre of gravity

contact disc

κ8 = −0.93

κ8 = −0.71

κ8 = −0.47

κ8 = −0.27

(a) front view

centre of gravity

contact disc

Fs

κ8 = −0.47

(b) lateral view

Fig. 4.23: specimen with four positions of the lever arm of the spring with respect tocentre of gravity, i.e. four values of parameter κ8. (a) front view. (b) lateral view with theline between centre of gravity and the mounting hole for κ8 = −0.47 being vertical

85


This refers to the critical coefficient as defined in equation (4.2.13):

µc =1

κ3 (1 + κ8). (4.3.3)

Thus, the lower the value of κ8, the higher the critical coefficient. However, this will alsodecrease the maximal possible reduction. For a mutual verification, the extended modelis used, which is described in section 5.2.3 and appendix B.4. In comparison to theexperiments shown in section 4.3.1 a stiffer material for the connection element is usedand the mass of the specimen is increased to lower the influence of unwanted side-effects.With this, the maximal reduction parameter range is identified as listed in Tab. 4.3.2.Theoretically, this combination gives a maximal reduction of about 50%.

Tab. 4.3.2: maximum parameter range of the experimental set

set κ1 κ2 κ3 κ4 µ κ7

5 ≈ 2 1.33 1.51 2 · 10−4 0.75 1.77

Figure 4.24 shows a contour plot of the effective coefficient of friction for 1600 combi-nations of v0 and κ8. Both, higher values of v0 and κ8 increase the reduction. Thus, ahigher rotational moment of the spring force with respect to the contact spots leads tohigher reduction. Additionally, the yellow area shows the limit range which is delimitedby the black line. Here no reduction occurs at all. Thus, one can conclude that thereexists a limit velocity that has to be exceeded in order to induce the reduction effect.And that this limit velocity depends on the lever arm of the spring.

20 40 60−1

−0.8

−0.6

−0.4

no effect


κ8

0.4

0.6

0.8

1

µe

Fig. 4.24: surface plot of the effectivecoefficient of friction µe as a function ofv0 and κ8. Simulation model

κ8 = −0.27

κ8 = −0.47

κ8 = −0.71

κ8 = −0.93

10 20 30 40 500

0.2

0.4

0.6

0.8

1


tim

eof

inst

abilit

yt i

s/T

Fig. 4.25: area plot of time portion ofthe instability tis/T as a function of basevelocity v0 for different κ8. Experiment

Figure 4.26 gives a comparison of the theoretical values, which are symbolized by thecircles, and the experimental results, which are denoted by the error-bars and marks.

86


Although there occurs a certain deviation between theory and experiments, the overalltrend is similar. A certain velocity limit exists that has to be exceeded for the reductioneffect to occur. For κ8 = −0.93 and κ8 = −0.71 this velocity threshold is about v0 =40 mm/s respectively v0 = 25 mm/s. This agrees with the simulation as shown by thecorresponding pink and blue circles. For κ8 = −0.47 and κ8 = −0.27 the limit velocityis approximately v0 = 15 mm/s. This does not coincide with the simulation, where nolimiting velocity exists for these values of κ8. However, in the experiments the reductionincreases with increasing velocity once the limit velocity is exceeded, what agrees withthe simulation. It turns out that for every κ8 examined, the specific maximal reductionis higher in the experiments. For κ8 = −0.93 the reduction is 52%, for κ8 = −0.71 it is48% and for κ8 = −0.47 it is 60% in the experiments. The maximal reduction is 73%and occurs for κ8 = −0.27.

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1


effec

tive

fric

tion

µe

κ8 = −0.27κ8 = −0.47κ8 = −0.71κ8 = −0.93

Fig. 4.26: effective coefficient of friction µe as a function of base velocity v0 for differentκ8. Simulation (circles) and experiment (error-bars and marks) show that a velocity limithas to be exceeded to induce the friction minimizing effect

In addition, Fig. 4.25 gives area plots of the time portion of instability tis/T for the fourcases of κ8. The time of instability is determined on basis of the revolving variance asdefined in equation (4.3.1). It shows that tis increases with the base velocity v0 and leverarm ratio κ8. Thus, the higher the rotational moment of the spring force of the base withrespect to the contacts, the higher is the relative time of instability. In addition, thevelocity threshold for the instability to occur, i.e. the velocity for which tis ≈ 1 applies,is lower for a higher rotational moment. Comparison of Fig. 4.25 and Fig. 4.26 showsthat the reduction effect also highly coincides with increased time portion of instability.The experimental results confirm the results of the DoE as shown in Fig. 4.7, where thereduction increases with increasing κ8 and increasing velocity, i.e. increasing parameterκ6. As stated in section 4.2.2, the reduction effect is caused by self-excited verticaloscillations of the rigid body. These oscillations increase with an increasing moment ofthe spring force with respect to the contacts what increases the coupling between the

87


rotational moment and the friction forces. In addition, the oscillations increase with thevelocity as this increases the kinetic energy that is transferred to the vertical motionthrough the coupling effect. The effect of decreasing kinetic friction with increasingvelocity was experienced in many experiments [77, 78, 79] where an overview is given forexample in [9]. The same effect also occurs in theory as in the slip wave model of Adams[50] or in the rigid body model of Martins et al. [10]. However, the deviations betweenexperiment and model are relatively high. One reason for this might be a deviationbetween the real parameters of the experiment and the theoretical values. Another onemight be that the model does not capture the true contact mechanics. Section 5.2.3gives a detailed discussion of the deviations between theory and experiment.

4.4 Summary

On basis of important experimental [39, 40] and theoretical works [10, 47] a simulationmodel is introduced, in order to examine the influence of the system dynamics on slidingfriction. The model captures known basic effects as the coupling between normal andtangential motion and adds features as the spatial variation of stick and slip zones. Intheory, the frictional resistance almost vanishes when a certain parameter range is met.If so, self-excited vibrations in the vertical direction occur and the system moves in acharacteristic dynamic mode. The different mechanisms that are responsible for thereduction can be summarized as:

1. self-excited oscillations in the vertical direction caused by the coupling effect

2. strong correlation between low or even zero contact forces and forward motion

3. characteristic alternation between storage and motion phase

In the reduction mode the system moves in such a way that the forward motion ishighly synchronized with the vertical vibrations and thus normal forces in the contact.It moves forward at low normal forces or even released contacts and almost stops at highnormal forces. This dynamic mode is referred to as micro-walking. The amplitudes ofthe vertical oscillations can be rather small, i.e. they are from the order of the staticvertical displacement, and can thus be characterized as microscopic. This means that asignificant reduction can be accompanied with apparently smooth sliding of the system,i.e. the oscillation can remain undetected in a real system.Both, simulations and experiments, show that the reduction requires a sufficient speedand high moment of excitation with respect to the contact spots to enhance the kine-matic coupling and the self-excited vibrations. Another important criterion for the effectto occur is a similarity of the excitation and contact stiffness, meaning that there is amedium guidance of the base motion. The experiments confirm the theoretical resultswith the maximal reduction being 73%. However, there is a discrepancy between simu-lation and experiment as the final results differ slightly indicating that further researchis needed. In this, attempts should be made to achieve the theoretical maximal rangewith a reduction of 98 %, i.e. with an almost vanishing macroscopic frictional resistance.

88

Chapter 5

Experimental Analysis

Two tribological experimental rigs have been built in order to examine the theoreticalresults. Firstly, the rolling body test rig. It is used to investigate the effects of shakedownand ratcheting in incomplete contacts and is described in section 5.1. Secondly, themicro-walking machine test rig which is used to measure the influence of system dynamicson sliding friction. This experimental rig is introduced in section 5.2. The computersand measurement devices of the Institute of Mechanics have been used for measurementand control of the experiments. The measurement results in the present work are givenwith confidence interval on the confidence level 68.3%, as described in appendix A.1.1.

5.1 Experimental Rig for the Rolling Body Effects

The experimental rig sketched in Fig. 5.1 is used to investigate the effect of oscillatingrolling. It is mounted on a vibration-isolated table in order to minimize the influenceof external dynamic interferences. The experimental rig consists of a drive unit withjoint kinematics, rolling body (1) and a movable substrate (2), which is mounted on afriction reduced cross-roller table. The tangential force is applied through a weight potwhich is connected to the substrate via a string. This enables an efficient and precisecontrol of FT which is the result of the force of gravity of the pot G = mT g reduced bythe resistance force of the cross-roller table, i.e. FT = mT g − FW . The resistance forceis measured to be FW = 0.1 N. The force of gravity of the sphere acts as the normalforce FN , which is measured at the centre of the contact using a load cell. The rollingmotion is generated by the linear drive and the joint kinematics. In theory, the fulcrumof the lever arms is on the same level as the contact point of the rolling body, see alsosection 5.1.5. Consequently, a lateral motion of the linear drive with amplitude W leadsto a rotation of the lever arm with an angle of rotation:

ϕ ≈(

W

R

)

, (5.1.1)

where R denotes the radius of the rolling body. This leads to a lateral motion of thebearing points xm und xo shown in Fig. 5.1. Assuming small amplitudes ϕ ≪ 1 the

89

Chapter 5. Experimental Analysis

lateral motion results to:

xm = W and xo = 2W , (5.1.2)

what leads to a rolling motion of the body in which the actual point of contact corre-sponds to the instantaneous centre of motion.

g

lever

linear drive

cross-roller table

weight, mT

laservibrometer

roller (1)

substrate (2)

xo

xm

µ

R

U

ϕ

Fig. 5.1: experimental rig for the oscillating rolling: rolling body (1), kinematic joints andmovable substrate (2)

U

2Wsensor

controller

controller

C-863

OFV 503

OFV 5000PC

data boardNI-RXI-5922

Fig. 5.2: experimental rig for the oscillating rolling with measurement chain consisting oflaservibrometer, controller and PC

In order to keep the deviation from theoretical, ideal rolling motion as small as possible,an accurate adjustment of the geometry is needed. This is achieved using spacers betweenthe substrate and the cross-roller table and spring washers on the upper bearing points.The entire experimental rig including the measurement chain is depicted in Fig. 5.2. Asdescribed previously, the rolling motion is generated by a high precision linear drive M403.02-DG/M 405-DG which is controlled using a C-863 controller. Both devices are

90


made by Physik Instrumente GmbH & Co. KG. The displacement of the substrate U ismeasured by a laservibrometer. It consists of the sensor unit OFV 503 and the controllerFV 5000 both made by Polytec GmbH. The measurement resolution in the experimentsis 80 nm. The electrical output signals of the controller are fed to the PC using a 24-bitdata acquisition card NI PXI-5922 made by National Instruments Corporation.

5.1.1 Parallel Alignment of Force and Oscillation

Figure 5.3 shows the experimental set-up for the parallel case, including the vibration-isolated table. Here the rolling body (1) is a massive steel sphere and the substrate(2) consists of silicone rubber with a thickness of 8 mm. Both, the Young’s modulusof the substrate E2 and the coefficient of friction between sphere and substrate µ aredetermined using the least square method (LS), as described in appendix A.1.2. In this,the static displacement Ustat as a function of FT is measured firstly and the accordingstatic displacement for different combinations of the parameters E2 and µ is computedusing equation (2.1.18). Secondly, the parameters for which the sum of the squareddeviations between experiment and theory is the smallest are identified. For the parallelset-up, this gives E2 = 5.2 MPa and µ = 0.57. There are two reasons for this procedure.Firstly, the experimental setup does not allow a sufficiently accurate direct measurementof µ. Secondly, only mean values of the Shore-A hardness without tolerance ranges areknown. If the given mean value for the Shore-A hardness is used, the Young’s modulusresults to E2 ≈ 5 MPa according to Boussinesq, respectively E2 ≈ 6 MPa according toKunz and Studer where both values are taken from [98]. This matches the values gainedusing the mean square method. In addition, the optical measurements as explained insection 5.1.5 give almost the same contact radii as the theory taking into account thefitted values. Table 5.1.1 gives all important parameters of the parallel set-up.

0

5

10 [cm]

Fig. 5.3: rolling body experimental rig with sphere and parallel alignment of tangentialforce FT and oscillating rolling W

91


Tab. 5.1.1: parameters of the rolling body experimental rig with sphere and parallelalignment of force FT and direction of the rolling W

parameter symbol value unit

radius R 40 mmcoefficient of friction (LS) µ 0.57 1

Young’s modules (LS) E1/E2 206 · 103/5.2 MPaPoisson-ratios ν1/ν2 0.3/0.5 1normal force FN 21.1 N

contact radius a 4.56 mmindentation depth d 0.52 mm

5.1.2 Perpendicular Alignment of Force and Oscillation

In this case, the rolling is aligned perpendicular to the tangential force as shown inFig. 5.4. Again, a solid steel sphere acts as the rolling body (1) and silicon rubber withthickness 8 mm is used for the substrate (2). Likewise, the parameters µ and E2 aredetermined with the least square method: E2 = 4.35 MPa and µ = 0.93. The differenceto the parallel case is because another sort of silicone from another supplier is used.Again, the optical measurements as explained in section 5.1.5 show a good agreementwith the theory taking into account the fitted values. Table 5.1.2 provides a summaryof all important parameters for the perpendicular alignment.

0

5

10 [cm]

(a) lateral view

0

5

10 [cm]

(b) top view

Fig. 5.4: rolling body experimental rig with sphere an perpendicular alignment of tangen-tial force FT and oscillation W . (a) lateral view and (b) top view

92


Tab. 5.1.2: parameters of the rolling body experimental rig with sphere and perpendicularalignment of force FT and direction of the rolling W


radius R 40 mmcoefficient of friction (LS) µ 0.93 1

Young’s modules (LS) E1/E2 206 · 103/4.35 MPaPoisson-ratios ν1/ν2 0.3/0.5 1normal force FN 21.19 N

contact radius a 4.8 mmindentation depth d 0.58 mm

5.1.3 Cylindrical Roller and Parallel Alignment

In this case, a cylindrical roller with radius 40 mm and length 80 mm is used as therolling body (1) and silicone rubber with 5 mm thickness is used for the substrate (2).As shown in Fig. 5.5 (a), rolling motion and tangential force are aligned parallel to eachother. The cylinder has a 1 mm chamfer and almost flushes with the substrate. Thisconfiguration is chosen in order to minimize the influence of the edges of the contact onthe pressure distribution, as discussed in detail in section 5.1.5. Again, another sort ofsilicone rubber is used. With the least square method the parameters result to µ = 0.79and E2 = 3.6 MPa. In this case the results of the three dimensional simulation areused for comparison. Again, the fitted values show good agreement with the opticalmeasurements. Other important parameters are listed in Tab. 5.1.3.

0

5

10 [cm]

(a) lateral view

0

5

10 [cm]

(b) front view

Fig. 5.5: (a) rolling body experimental rig with cylindrical roller an parallel alignment oftangential force FT and rolling amplitude W . (b) cylindrical roller almost flushes with thesubstrate

93


Tab. 5.1.3: parameters of the rolling body experimental rig with cylindrical roller andparallel alignment of force FT and direction of the rolling W


radius R 40 mmlength L 80 mm

coefficient of friction (LS) µ 0.79 1Young’s modules (LS) E1/E2 206 · 103/3.6 MPa

Poisson-ratios ν1/ν2 0.3/0.5 1normal force FN 32.3 Ncontact width 2a 4.2 mm

indentation depth d 0.26 mm

5.1.4 Experimental Procedures

In order to increase the reproducibility, it is briefly described how the experiments areconducted and a few time responses of measured values are shown. For the shakedowncase the final experimental results are given in the sections 3.2.2, 3.4.2 and 3.5.3. Forthe ratcheting case the final experimental results are given in the sections 3.3, 3.4.3 and3.5.4. The parameters of influence fT and w as well as the displacement u are normalizedas described in equation (3.1.4) for the spherical rollers respectively equation (3.5.13)for the cylindrical roller.

Shakedown for Spherical and Cylindrical Rollers The procedure to measure theshakedown displacement usd of the substrate is as follows:

1. placing of the rolling body on the substrate

2. application of the tangential force fT using the weight pot

3. wait for 60 − 90 s until static equilibrium is reached

4. oscillating rolling for n = 12 periods with specific subcritical amplitude w < wlim

In order to study a particular combination of parameters three to five iterations, i.e.measuring runs, are performed before the experimental set-up is changed. Figure 5.6shows the time response of the displacement of the substrate for fT = 0.19 and differentw in case of w ⊥ fT . The displacement stops after several periods and a safe shakedownoccurs. The small oscillations around the shakedown displacements are caused by slightgeometrical deviations, as explained in section 5.1.5. The last value of the displacementwas chosen as the resulting displacement after shakedown. The roller is then in its initialupright position. Examples of time responses for w ‖ fT and for the cylindrical rollerare not given here, as they are very much the same, except for the final displacement.

94


Shakedown Limits for Spherical Rollers Figure 5.7 gives the displacement forfT = 0.64 and w ⊥ fT for the limit case and three experimental runs. These experimentsare used to determine the maximal amplitude wlim. Therefore, starting from a safeshakedown state, the amplitude is increased stepwise W = W + ∆W until ratchetingoccurred. The asterisks delimit the intervals in which the amplitude is held constant.The complete procedure to identify the maximal amplitude wlim is as follows:


2. application of the tangential force fT using the weight pot


4. oscillating rolling for n = 5 − 10 periods and subcritical amplitude w < wlim

5. increasing of the amplitude W = W + ∆W

6. repeating of step 4. and 5. until ratcheting occurs

The incremental increases of the amplitudes are chosen as:

w ‖ fT ⇒ ∆W = 0.1 mm , (5.1.3)

w ⊥ fT ⇒ ∆W = 0.05 mm . (5.1.4)

The last amplitude before ratcheting is identified as the maximal amplitude wlim andthe result is the mean of three measurement runs. The time responses for the parallelcase w ‖ fT are not given here, as they match with the perpendicular case except forthe exact values.

0 1 2 3 40

0.1

0.2

0.3

time t [min]

dis

pla

cem

ent

u

w = 0.23w = 0.48w = 0.71

Fig. 5.6: displacement u for differentoscillation amplitudes w and fT = 0.19in case of w ⊥ fT and shakedown

0 50 100 150

0.4

0.6

0.8

1

ustat

shakedown

ratcheting

time t [min]

dis

pla

cem

ent

u

run-1run-2run-3

Fig. 5.7: three measurement runs forthe displacement u for fT = 0.64 andstepwise increase of w in case of w ⊥ fT

for determination of wlim

95


Shakedown Limits for Cylindrical Rollers The measurement procedure for deter-mination of the maximal amplitude wlim in case of the cylindrical roller resembles theone for the spherical rollers:


2. application of tangential force fT using the weight pot


4. oscillating rolling for n = 60 periods and sub-critical amplitude w < wlim

5. increase amplitude W = W + ∆W with ∆W = 0.05 mm

6. repeat step 4. and 5. until ratcheting occurs

Unfortunately, the distinction between shakedown and ratcheting is not as clear as forthe spherical case even for a highly increased number of periods, i.e. 60 instead of 10.Figure 5.8 shows the displacement for fT = 0.42 and a stepwise increase of w for threeexperimental runs. As can be seen, there occur regions of clear shakedown and regions ofclear ratcheting with high incremental displacement per period. Here, the displacementfor fT = 0.42 and a stepwise increase of w is shown for three experimental runs. Theasterisks delimit the intervals in which the amplitude is held constant. It shows that thetransition region is ambiguous what makes it hard to identify the maximal amplitude.

0 10 20 30 40

0.5

1

clear ratcheting

clear shakedown

(1)(2)

time t [min]

dis

pla

cem

ent

u

run-1run-2run-3

Fig. 5.8: displacement u for fT = 0.42and stepwise increase of w in case of thecylinder for determination of wlim

0 0.2 0.4 0.6 0.8 10

0.20.40.60.8

1

u,u

p

u(1)

up(1)

0 0.2 0.4 0.6 0.8 10

0.20.40.60.8

1

time t

u,u

p

u(2)

up(2)

Fig. 5.9: conditioned displacement uand polynomial fit up for the intervals (1)and (2) in Fig. 5.8

In order to give a distinct and comprehensible criteria, the displacement of the substrateu for a specific amplitude w is approximated with a second order polynomial fit:

up = p2 (w) t2 + p1 (w) t + p0 (w) . (5.1.5)

96


Here, u, up and t denote the conditioned and normalized displacement and time suchthat in the interval the displacement starts with 0 and reaches 1 at the end:

u(

t = 0)

= 0, u(

t = 1)

= 1 . (5.1.6)

An example for the conditioned displacement as well as the polynomial approximationis given in Fig. 5.9. These refer to the intervals (1) and (2) of Fig. 5.8. It is assumedthat the system still approaches a state of shakedown as long as the curvature of thepolynomial fit (5.1.5) is negative. Otherwise ratcheting must have started:

∂2up

∂t2< 0 ⇒ shakedown , (5.1.7)

∂2up

∂t2> 0 ⇒ ratcheting . (5.1.8)

The last shakedown amplitude is then identified as the maximal shakedown amplitude.For the examples given in Fig. 5.9 the curvatures yield −0.65 for interval (1) and 0.13 forinterval (2). Thus, one can conclude that the system still shakes down for w in interval(1) and ratcheting has started for w in interval (2). This is done for all the measurementruns.Finally, the maximal amplitude is computed as the mean value of three measurementruns. Due to the described processes the maximal displacement cannot be read directlyas the system has still not reached the final shakedown state. Therefore, an artificialcriterion for determining the maximal displacement is defined. The displacement ofthe last shakedown interval Usd and the first ratcheting interval Urt are approximatedlinearly:

Usd (t) = p11t + p01 and Urt (t) = p12t + p02 . (5.1.9)

The intersection of the two lines Usd (t) = Urt (t) is assumed to give the maximal dis-placement:

Ulim =p02 − p01

p11 − p12p11 + p01 . (5.1.10)

Although the maximal displacements match the theoretical values relatively good, seeFig. 3.45 in section 3.5.3, the deviations of the maximal amplitudes are relatively high,see Fig. 3.44 in section 3.5.3. For the maximal amplitudes wlim the mean deviation isapproximately 16%. Thus the maximal force fT,lim in the experiments is 16 % higherthan the theoretical prediction. Due to the strong agreement between theory and sim-ulation, see again Fig. 3.44 in section 3.5.3, one can conclude that the experiment issomehow faulty. Potential causes are:

• an incorrect criterion for the determination of the maximal amplitude

• deviations in the alignment of tangential force and rolling amplitude

• deviations in the pressure distribution, as discussed in section 5.1.5

• erroneous determination of the parameters of the experimental rig

97


Ratcheting for Spherical and Cylindrical Rollers The measurement procedurefor determining the incremental displacement ∆u is the same in all three cases that areconsidered:


2. application of tangential force fT using the weight pot


4. oscillating rolling for n = 10 − 20 periods and supercritical amplitude w > wlim

In order to study a particular combination of parameters three to five iterations, i.e.measuring runs, are performed before the experimental set-up is changed.

5.1.5 Analysis of Deviations

Dynamic Influences In order to estimate the dynamic influences of the test rig, theeigenfrequencies of the system are determined. For this purpose, the rolling body andthe main board of the joint kinematics are assumed to be rigid and only three degreesof freedom are considered: translation and rotation of the roller and translation of thesubstrate. The remaining components are modelled as linear elastic springs. With thevalues of Tab. 5.1.1 the first eigenfrequency [99] for the spherical roller and the parallelalignment of force and oscillation results to f1 = 22 Hz. The highest excitation frequencyoccurs for the smallest amplitude W = 1

4a. With a drive speed of W = 1 m/s the highestexcitation frequency results to f0 = 0.21 Hz and is thus two orders of magnitude lessthan f1. The excitation and the displacement of the substrate can thus be assumed tobe quasi-static in relation to the natural oscillations of the system. In turn, its dynamicinfluences are neglected.Also for the perpendicular alignment the eigenfrequencies are estimated using a simplifiedsystem with rigid body degrees of freedom only. With the values of Tab. 5.1.2, the firsteigenfrequency results to f1 = 19 Hz which also exceeds the highest excitation frequencyof f0 = 0.21 Hz by two orders of magnitude. The difference between the two settings iscaused by the fact that the bending stiffness of the lever arms is lower than their tensilestiffness. Again, the dynamic influences of the system are neglected. For the cylindricalroller, the eigenfrequencies are assumed to be comparable to the settings with sphericalrollers. Thus the excitation and the displacement of the substrate are again assumed tobe quasi-static in relation to the natural oscillations.

Kinematic Deviations Figure 5.10 depicts the rigid body displacement of the spher-ical rolling contact for w ‖ fT in case of shakedown. As one can see, some slightoscillations occur in the shakedown state. These are caused by small geometrical de-viations of the experimental setting. The model is such that the main bearing of thelever kinematics is on the same level as the lowest point of the rigid sphere, i.e. thecontact point. Thus, the instantaneous centre of motion of the sphere coincides with its

98


lowest point. However, in the experiment occurs a small deviation ∆h such that the trueinstantaneous centre is on a higher level, as depicted in Fig. 5.11. During rolling, thiswill induce a small displacement of the substrate ∆U which is a function of the rotationϕ ≈ W

R around the instantaneous centre of motion:

∆U =W

R∆h . (5.1.11)

This effect causes slight oscillations of the displacement around the shakedown valueas can be seen in Fig. 5.10. These oscillations also occurred in the unloaded case, i.e.fT = 0. For the parallel case and fT = 0 and w = 0.75 the amplitude of the oscillationis measured:

∆U = (1.55 ± 0.13) µm ⇒ ∆h = (62.17 ± 5.38) µm . (5.1.12)

One can assume that these oscillations are from the same order for all the cases that areconsidered. Although these geometric deviations might have a small influence on thecontact area, this effect can be neglected. The displacement u when the roller is in itsupright position is taken as the experimental result usd for the shakedown case.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

ustat

usd

periods n

dis

pla

cem

ent

u

w = 0.24w = 0.50w = 0.75

Fig. 5.10: displacement u for differentoscillation amplitudes w and fT = 0.24in case of shakedown for w ‖ fT (experi-ment)

leverroller

substrate

∆h

∆U

ϕ

Fig. 5.11: height difference betweenbearing and contact point ∆h causes de-viation of the displacement ∆U

Half-Space Assumption In the experiments, a rubber substrate is used that is gluedto a metal plate which again is mounted on the cross roller table. It is to be discussedto which extent the finite layer thickness of the rubber b contradicts the theoretical as-sumption of a semi-infinite substrate. The half-space model assumes that all boundariesof the semi-infinite bodies are sufficiently distant from the contact, so that the relativedisplacements in the contact are independent of the remote boundary conditions [52].The elastic deformations decrease with 1/r, with r being the distance to the contactarea, and the stresses and strains decrease in proportion to 1/r2. Therefore, the elasticenergy of the deformation is concentrated in a volume with linear dimensions of the

99


order of the contact width Dc [65], as shown in Fig. 5.12. Considering this, the layerthickness is of great influence for the contact stresses and the contact area. In case thatb ≫ Dc, the half-space assumption is not violated and the contact stresses are given bythe Hertz theory [56]. In case that b < Dc the contact area is reduced and the maximumpressure is increased in comparison with the Hertzian solution [100].

p (x, y)

Dc

r ≈ Dc b

roller

layer

Fig. 5.12: model of the contact betweenroller and elastic layer of thickness b. De-formation is concentrated in a volumewith linear dimensions of the order of thecontact width Dc (shaded area)

[mm]

Dc

Fig. 5.13: optical measurement of thecontact area in case of the sphericalroller. The diameter is approximatelyDc ≈ 9 mm

For the experiments, the ratios of the layer thickness b and the contact length Dc arecalculated as:

(

b

Dc

)

sphere=

8 mm

9 mm≈ 0.9 and

(

b

Dc

)

cylinder=

5 mm

4.2 mm≈ 1.2 . (5.1.13)

The layer thickness is thus of comparable size as the contact length Dc. For frictionalcontacts incorporating elastic layers as in the experiments presented here, the contactcan assumed to be effectively Hertzian for b ≥ Dc [101, 52]. Thus, one can concludethat the experiments are sufficiently covered by the Hertzian theory and satisfy the half-space assumption. This is also supported by optical measurements of the contact radii.In these, the rollers are covered with stamping ink before indentation. After removingthe roller from the substrate, a coloured imprint remains at the surface that indicatesthe size of the contact area, as shown in Fig. 5.13. The measured radii agree well withthe theoretical values computed with the Hertz theory.

Boundary of the Cylindrical Contact The model of the cylindrical rolling contactassumes the stress distribution to be constant in the y direction perpendicular to therolling. However, the finite length of the cylinder causes a much sever state of stressin the contact. Although the Hertzian theory predicts the pressure correctly over themajority of the contact, significant deviations occur at the boundaries [56], as shown inthe diagrams in Fig. 5.14. In case of two coincident sharp ends as shown in Fig. 5.14(a), the free end of the solids expand slightly what reduces the pressure as:

p′0 ≈

(

1 − ν2)

p0 . (5.1.14)

100


A sharp stress concentration occurs when the mating substrate extends beyond theroller, as in Fig. 5.14 (b). For a frictionless contact with β = 0, the pressure in the closeneighbourhood of the edge will vary as y−0.23 [102]. This problem is typically avoidedby barrelling the rollers as depicted in Fig. 5.14 (c). Here, the pressure increases slightlyand then relaxes, resulting in a dog-bone shape of the contact area. Provided that theradius at the ends is significantly larger than the contact width, both bodies can beregarded as half-spaces [56]. Kunert gives the profile correction for the cylinder, that isneeded to leave unchanged the elliptical pressure in the rolling direction but to achievea constant pressure in the perpendicular direction [89]. However, such a profile is verydifficult to manufacture and is correct only at the design load.

x

y

ya

a

p0 (y)p′

0

(a)

x

y

ya

a

p0 (y)

(b)

x

y

a

a

p0 (y)

(c)

Fig. 5.14: boundary effects of the cylindrical roller. (a) two coincident sharp ends, (b) onesharp end and (c) rounded end, taken from [56]

Considering these relations and practical restrictions of the measurement set-up, thetest rig is built as a mixture of type (a) and (c). The cylinder has a 1 mm chamferand almost flushes with the substrate. In addition, the ratio of the length of the linecontact L and the contact width 2a is L/a ≈ 20. Although these characteristics mightminimize the boundary effects, they cannot fully be neglected. The increased maximalpressure in case (c) might be one reason for the increased maximal load that is observedin the experiments in comparison with the theoretical value, as shown in section 3.5.3in Fig. 3.44.

Viscoelasticity of Rubber The silicone rubber, which serves as the substrate in theexperiments, is assumed to be elastic in the considered parameter range. In fact therheology of rubber is characterized as viscoelastic, meaning that it exhibits both elasticand viscous characteristics. Mechanically, this is described by the time dependent shear

101


modulus G (t) function:

G (t) =1

2π

∫ ∞

−∞

1

ω

(

G′ sin (ωt) + G′ cos (ωt))

dω . (5.1.15)

Here G′ and G′′ denote the storage respectively the loss modulus and ω is the frequencyof the stress acting inside of the rubber [55]. The standard rheological model for rubberis a combination of a spring (G1) and a so called Maxwell element, which is a seriesconnection of another spring (G2) and a dash-pot (η), as shown in Fig. 5.15 (b). Usuallyfor rubber, G2 is much larger than G1.

FN

G (t)

(a) rubber substrate

FN

G1 G2

η

(b) standard model

Fig. 5.15: (a) contact of rubber substrate and indenter. (b) standard rheological modelfor rubber

As this standard model is a parallel connection, the corresponding modules are given as:

G′ = G1 + G2(ωτr)2

1 + (ωτr)2 and G′′ = G2(ωτr)2

1 + (ωτr)2 , (5.1.16)

where τr = ηG2

denotes the relaxation time. For low frequencies, i.e. ω < G1/η, rubberbehaves almost elastic with G (t) ≈ G1. Likewise for very high frequencies, i.e. ω >G2/η, rubber also behaves almost elastic with G (t) ≈ G2 [55]. The viscous dissipationcan be neglected in both cases. In contrast, in the intermediate range, rubber behaveslike a viscous fluid doing periodic loading, i.e. it exhibits high dissipation. Now, for thesilicone rubber used in the experiments the viscosity is approximately η = 40 Pas [103]and the quasi-static shear-modulus is measured G1 = 1.7 MPa as listed in Tab. 5.1.1.Given the speed of the linear drive of W = 1 mm/s, the highest excitation frequencyresults to f0 = 0.21 Hz what yields:

ω

G1/η=

2π · 0.21 Hz · 40 Pas

1.7 MPa≈ 3 · 10−5 ⇒ G (t) ≈ G1 . (5.1.17)

Thus, one can assume the rubber to behave elastic in the examined parameter range.

102


5.2 Experimental Rig for the Micro-Walking Effect

The experimental set-up shown in Fig. 5.16 is used to investigate the micro-walking effect.Most important components are the specimen and the substrate. The drive moves thesubstrate with velocity v0 relative to the specimen. Thus, the system corresponds to thewell-known moving belt model.

v0

Fs Fs force-sensorTD-112/KD 24s

driveZwick Z1.0

controller

Zwick BX 1

amplifier

HBM MC 55PC

data board

NI-PCI-6221

Fig. 5.16: experimental rig for the micro-walking effect with measurement chain consistingof force sensor, amplifier, data board and PC

The experimental rig is powered by an electro-mechanical testing actuator made byZwick GmbH & Co. KG whose good ganging properties reduce unwanted dynamic in-teractions. The actuator is controlled using the controller BX-1. Straight rods act asthe spring between specimen and base. The spring force Fs is measured at the base ofthe spring using a S-force sensor. Different sensors are used, namely the sensor TD-112by Yuyao Tongda Scales Co., Ltd. and the sensor KD24s by ME measurement systemsGmbH. The latter one has smaller dimensions and allows a larger adjustment range ofthe height of the base. For amplification of the sensor signal, the amplifier system MGCof the company HBM is used, which essentially consists of the amplifier HBM MC 55.The measured signals are fed to the PC via a measuring system with 16-bit data acquisi-tion board, namely NI PCI-6221 by National Instruments Corporation, which also servesto control the drive. The sampling frequency used to measure the spring force Fs is cho-sen to be fsamp = 4000 Hz. This satisfies the Whittaker–Nyquist–Kotelnikov–Shannontheorem [104] as:

fsamp = 4000 Hz > 2 × Ωexp

2π≈ 760 Hz , (5.2.1)

where Ωexp is determined from the frequency spectrum of the time response of Fs asshown in section 5.2.3. Finally, the effective coefficient of friction µe is computed as:

µe =〈Fs (t)〉√

2µmg. (5.2.2)

103


5.2.1 Parameters of the Experimental Rig

The materials and the exact dimensions of the specimen and the substrate are chosensuch that the dimensionless parameter combinations, which are determined with thenumerical model as described in section 4.2.1 and section 4.2.2, are met as closely aspossible. However, experimental feasibility leads to certain limitations as discussed be-low. The specimen has an hourglass shape as shown in Fig. 5.17 (a). It consists of asteel centrepiece and two contact discs made of polypropylene (EP P = 1.4 GPa). Thisconfiguration enables a quick adjustment of the geometry and thus inertia properties.The substrate consists of smooth window glass (EGS = 90 GPa) and has a prism shapeas shown in Fig. 5.17 (b) and Fig. 5.18 (a). On the one hand, this configuration en-hances the stability and parallelism of the motion of the specimen. On the other hand,the contact configuration of the experiment differs slightly from the initial theoreticalmodel introduced in section 4.1.3. Thus, for the comparison in section 4.3.2 an extendedmodel as described in sections 5.2.3 and appendix B.4 is used. Other dimensions andparameters of the experimental rig and the different combinations for set 1-5 are givenin Tab. 5.2.1. A detailed discussion of how these parameters are determined is givenbelow. In addition a deviation analysis is given in section 5.2.3.

v0D

Db

l1 l2

Rc

ls

(a) lateral view

µ

(b) front view

Fig. 5.17: (a) lateral view of the hourglass-shaped specimen with dimensions. (b) frontview of the specimen with prism-shaped substrate

0 5 10 [cm]

(a) experimental rig

0 10 20 [mm]

(b) mount of the spring

Fig. 5.18: (a) experimental rig for the micro-walking effect consisting of specimen andprism-shaped substrate. (b) mount of the spring at the S-force sensor with spring rod

104


Tab. 5.2.1: parameters of the micro-walking experimental rig for the different parametercombinations set 1-5. The numbers in brackets gives the corresponding set


radius Rc 5 mmcontact disc diameter D 60 mm

body diameter Db 30 (1, 2, 3, 4) /50 (5) mmmodel half-height a 21.2 mm

length of centre part l1 18 (1, 3, 5) /40 (2, 4) mmmodel half-width b 14 (1, 3, 5) /25 (2, 4) mmlength of side-disc l2 5 mmmass of specimen m 201 (1, 3) /327 (2, 4) /490 (5) g

length of spring-rod ls 100 mmfriction coefficient µ 0.79 (1, 2) /0.68 (3, 4) /0.75 (5) 1Young’s modules EP P /EGS 1.4/90 GPa

Poisson-ratios νP P /νGS 0.4/0.3 1contact diameter Dc 1.8 mm

Contact Stiffness and Contact Diameter The contacts between specimen andsubstrate are modelled as linear springs with normal and tangential stiffness:

kz = E∗Dc and kx = G∗Dc . (5.2.3)

The effective modules E∗ and G∗ directly depend on the materials in contact, whereasthe contact diameter Dc is a function of the actual contact configuration as sketched inFig. 5.19 (a). Hence, it depends on the state of motion of the system. In the following,different ways for the estimation of Dc are presented.

1) If the contacts are assumed to be of the Hertzian type with an average normalforce FN = 1

2√

2mg acting on each single contact the contact width yields:

Dc =

(

6FN Rc

E∗

)

≈ 0.4 mm . (5.2.4)

One drawback of this assumption is that in fact the normal force and thus Dc

change in time. In addition, it remains unclear if the contact model is chosencorrectly.

2) Due to wear, the shape of the contacts will soon not resemble a Hertzian contactbut a flattened ellipse with a more or less constant effective diameter as shown inFig. 5.19 (b). Here one wear spot in the contact zone is shown. A rather crudeoptical estimate of the conjugate diameters of the wear spots gives:

Dc = 2√

ab ≈ 2 mm . (5.2.5)

105


But again, the question remains whether the wear spots really correspond to thevisible contact zones.

3) Another way to estimate the contact stiffness is to compare the characteristicfrequencies of the spring forces of the experiment and the simulation as discussed insection 5.2.3. Assuming that the simulation maps the reality sufficiently accurate,the characteristic frequencies must match. This gives the characteristic time τ asdefined in equations (4.1.11) and (5.2.35) and one can compute the contact stiffnessas:

kx =m

τ2. (5.2.6)

The contact stiffness and diameter then yield:

kx = (2.2 ± 1.5) · 106 N/m and Dc = (1.8 ± 1.2) mm . (5.2.7)

The values stated in (5.2.7) are used for approximation and for the comparison of exper-iment and simulation in section 4.3.2 as these values resemble to the flattened ellipticalcontact.

FN

Dc E∗, G∗

(a) model of the contact

wear zone

specimen

(b) wear zone at the specimen

Fig. 5.19: (a) contact of elastic body and rigid substrate with contact diameter Dc as amodel of the contact zones of the between specimen. (b) contact disc with elliptical shapedwear zone

Spring Stiffness The parameter studies of the numerical model as described in sec-tion 4.2.1 show that the micro-walking effect occurs if for the dimensionless parametersκ1 and κ4 applies:

κ1 =ks

kx≈ 1 and κ4 =

mg

kxb≈ 10−4 . (5.2.8)

The use of a soft spring, e.g. ks = 103 N/m, would lead to experimental advantages asthis increases the difference between the eigenfrequencies of the specimen and those ofthe experimental rig, i.e. drive and base frame. However, in order to fulfil (5.2.8) the

106


half width b would result in completely impractical values, as can be shown for typicaldimensions of the experiment, e.g. m = 100 g, g = 9.81 m/s2:

ks = kx ⇒ b = 104 · mg

ks≈ 104 · 1 N

103 N/m= 10 m . (5.2.9)

Instead, stiff polypropylene is used for the contact discs, such that the tangential stiffnessresults to kx = 2 · 106 N/m and the half width to b = 10−2 m. In order to achieve aspring stiffness of ks ≈ kx, straight rods made of fibre-reinforced plastic respectivelypolystyrene are used. Their corresponding Young’s modules are measured on the basisof the DIN norm 178 [105] and result to:

EP S = (3.42 ± 0.02) GPa , (5.2.10)

ECF = (176.66 ± 2.02) GPa . (5.2.11)

The spring stiffness corresponds to the according longitudinal stiffness:

ks =EsAs

ls, (5.2.12)

where As is the cross sectional area of the rod. Initially, the spring stiffness result to

ks−P S = (2.42 ± 0.14) · 105 N/m , (5.2.13)

ks−SP = 60 N/m , (5.2.14)

ks−CF = (5.50 ± 0.63) · 106 N/m , (5.2.15)

where ks−SP denotes the spring stiffness of the spiral spring used in section 4.3.1. Thisvalue is simply taken from the corresponding data sheet, where the uncertainty of themeasurement is not given. However, the stiffness of the sensor used for determination ofthe spring force has to be taken into account. For the sensor KD24s the spring stiffnessis kKD24s = 2 · 107 N/m. As it is in series with the spring, the effective spring stiffnessks can be determined as :

ks =

(

1

ks+

1

kKD24s

)−1

, (5.2.16)

what finally yields1:

ks−P S = (2.38 ± 0.14) · 105 N/m , (5.2.17)

ks−CF = (4.34 ± 0.39) · 106 N/m . (5.2.18)

The stiffness of the sensor TD-112 is not known. As its nominal load is higher thanfor the KD24s it is expected to be stiffer. Thus, according to (5.2.16) its influence is

1The sensor can assumed to be rigid in comparison to the spiral spring, i.e. it does not influence thestiffness in this case.

107


lower. In the end, the high error of measurement for the contact stiffness discussed insection 5.2.3 leads to the following rather crude estimates of the parameters κ1:

κ1−P S =ks−P S

kx≈ 0.1 → used in set 1, 2 ,

κ1−SP =ks−SP

kx≈ 10−4 → used in set 3, 4 ,

κ1−CF =ks−CF

kx≈ 2 → used in set 5 .

(5.2.19)

Sets 1-4 are used in the analysis of section 4.3.1 and are tabulated in Tab. 4.3.1. Set 5is used in the comparison of experiment and simulation in section 4.3.2 and is describedin Tab. 4.3.2.

5.2.2 Experimental Procedures

In order to increase the reproducibility, it is briefly described how the experiments areconducted and a few time responses of the spring force are shown. The final experimen-tal results are given in sections 4.3.1 and 4.3.2. The experimental procedure used fordetection of the time response of the spring force is as follows:

1) cleaning of the glass substrate with degreaser

2) alignment of the base and specimen and attaching of the spring

3) measuring run with desired velocity v0

4) return run with velocity v0 = 15 mm/s

The running distance is 190 mm. In order to ensure an almost stationary contact configu-ration, i.e. elliptical wear patches at the contact zones, about 100 runs for the running-inprocess are conducted after every change of the experimental set-up. In order to studya particular combination of parameters 10 to 20 iterations, i.e. measuring runs, are per-formed before the experimental set-up is changed. The following figures depict cut-outsof the time response of the normalized spring force Fs for the parameter set 5 as de-scribed in section 4.3.2 in Tab. 4.3.2, i.e. the maximum parameter set of the experiment.The lever arm ratio is κ8 = −0.71. As shown in Fig. 5.20, where Fs for v0 = 8.33 mm/sis given, only small oscillations occur in case that the base velocity v0 falls below thelimit velocity. If the limit velocity is exceeded, instabilities with high frequency occur,as shown in Fig. 5.21 which gives Fs for v0 = 33.33 mm/s. These instabilities cause areduction of the efficient coefficient of friction. Comparison of the close ups in Fig. 5.22and Fig. 5.23 highlights the differences between the time responses.

108


0 0.2 0.4 0.6 0.8 1

−1

0

1

2

time t [s]

Fs(t

)/√

2µm

g

Fig. 5.20: time response of Fs for set 5with the parameters as in Tab. 4.3.2 andκ8 = −0.71 and v0 = 8.33 mm/s

0 0.2 0.4 0.6 0.8 1

−1

0

1

2

time t [s]

Fs(t

)/√

2µm

g

Fig. 5.21: time response of Fs for set 5with the parameters as in Tab. 4.3.2 andκ8 = −0.71 and v0 = 33.33 mm/s

0.7 0.72 0.74 0.76 0.78 0.80.9

0.95

1

1.05

1.1

time t [m]

Fs(t

)/√

2µm

g

Fig. 5.22: close up of the time responseof Fs for the parameters as in Tab. 4.3.2and κ8 = −0.71 and v0 = 8.33 mm/s

0.7 0.72 0.74 0.76 0.78 0.8

−1

0

1

2

time t [s]

Fs(t

)/√

2µm

g

Fig. 5.23: close up of the time responseof Fs for the parameters as in Tab. 4.3.2and κ8 = −0.71 and v0 = 33.33 mm/s

5.2.3 Analysis of Deviations

Contact Configuration The geometry and contact configuration of the experimentdiffers from the theoretical model. This is mainly caused by the inability to replicate thetwo-dimensionality of the theoretical model in practice. For instance, usage of a lateralguidance in the y-direction would induce an additional friction force to the system,especially since it would theoretically have to be free of bearing clearance. However, thethree dimensional experiment can be modelled as a two dimensional model that consistsof three dimensional contacts. This is possible, since for symmetrical initial conditions2,

2In this case the word symmetrical means that the rigid body is not rotated around the y- and z-direction and not displaced in the y-direction. This was tested by the author using a full 3-D modeltaking into account all 6 directions of motion.

109


the contact forces are symmetrical for all of the four contacts. The according equationsof motion are then the same as for the theoretical model that are given in equations(4.1.16)-(4.1.19). One difference relates to the contact forces. The contact plane isinclined since the substrate is prism shaped. This leads to an additional spring deflectionus as described in section B.4. There are four contacts spots each exhibiting its ownspring deflections. But as the model is symmetrical, the two contacts on the left andon the right obtain the same contact conditions. Thus, only one contact on the left andone on the right are considered and the corresponding contact forces are doubled:

fN1/2 = 2 · κ2

√2

2

(

un1/2 + us1/2

)

, (5.2.20)

fT 1/2 = 2 · ux1/2 . (5.2.21)

The equations of motion in the non-dimensional form yield:

x′s = κ6 , (5.2.22)

x′′ = −2ux1 − 2ux2 + κ1 (xs − x + κ8ϕ) , (5.2.23)

z′′ = −2κ2

√2

2(un1 + us1) − 2κ2

√2

2(un2 + us2) + κ4 , (5.2.24)

ϕ′′ =1

κ7

(

−2κ2κ3

√2

2(un1 + us1) (1 − ϕ) + 2κ2κ3

√2

2(un2 + us2) (1 + ϕ)

)

,

+1

κ7

(

−2ux1

(

κ23 + ϕ

)

− 2ux2

(

κ23 − ϕ

)

− κ1

(

κ23κ8 + ϕ

)

(xs − x + κ8ϕ))

. (5.2.25)

As a consequence of the inclined contact plane, the effective coefficient for the experi-mental model must be computed as:

µe =〈fs (t)〉√

2µκ4

. (5.2.26)

Hence, it corresponds to µe of the experiment as defined in equation (5.2.2):

µe =〈Fs (t)〉√

2µmg. (5.2.27)

Deviation of the Parameter Range Some of the parameters can simply be mea-sured geometrically. This is the case for the parameters κ3, κ7 and κ8. For these, thedeviation between the real values of the experiment and the calculated ones is expectedto be such small that it can be neglected. The microscopic friction µ can be determinedexperimentally using the test rig and base velocities below the threshold for the insta-bility. Different batches for the material of the contact disc result in different values forthe microscopic friction:

µ = 0.79 ± 0.01 → used in set 1, 2 ,µ = 0.68 ± 0.03 → used in set 3, 4 ,µ = 0.75 ± 0.03 → used in set 5 .

(5.2.28)

110


Sets 1-4 are used in the analysis of section 4.3.1 and are tabulated in Tab. 4.3.1. Set 5is used in the comparison of experiment and simulation in section 4.3.2 and is describedin Tab. 4.3.2. The value of κ2 can only be estimated as the real contact configurationremains unknown. This is not a problem, since the influence of κ2 is only minor as shownin section 4.2.1. Assuming an approximately circular contact area and considering theparameters of Tab. 5.2.1, the Hertzian contact theory yields the parameter κ2 as:

κ2 ≈ 1.33 . (5.2.29)

In contrast the remaining parameters κ1 and κ6 are of great influence as shown insection 4.2.1. Both depend on the unknown tangential contact stiffness kx. One way toestimate kx, is to determine the non-dimensional time:

τ =

√

m

kx⇒ kx =

m

τ2. (5.2.30)

For this purpose the time response of the normalized spring force of both, experimentand simulation, is determined. Afterwards, the corresponding frequency spectra Ss [99]are computed. For the simulation Ss is computed as:

Ss−sim =1

T

∫ T

0

fs (t)√2µκ4

exp (−iΩt)dt , (5.2.31)

and for the experiment it is computed as:

Ss−exp =1

T

∫ T

0

Fs (t)√2µmg

exp (−iΩt)dt . (5.2.32)

0 0.1 0.2 0.3 0.4 0.5−4

−2

0

2

4

6

time t [s]

Fs(t

)/√

2µm

g

Fig. 5.24: time response of the springforce Fs for the parameter set 5

0 100 200 300 4000

0.5

1

frequency Ωexp [s−1]

spec

trum

Ss−

exp

Fig. 5.25: frequency spectrum Ss−exp

of Fs for the for the parameter set 5

Figure 5.24 gives the time response of Fs of the experiment for the parameter set:

set 5: κ1−8 =(

2, 1.33, 1.51, 2 · 10−4, 0.75, 1.6 · 10−3, 1.77, −0.47)

. (5.2.33)

111


Figure 5.25 shows the corresponding frequency spectra Ss where the star denotes thecharacteristic frequency Ωexp. Assuming that the simulation maps the reality sufficientlyaccurate, the frequencies must match, i.e. Ωsim = Ωexp. In case of the simulation, thecharacteristic frequency Ωsim is given as the computed non-dimensional frequency ωsim

divided by the characteristic time τ :

Ωsim =ωsim

τ. (5.2.34)

Thus, the characteristic time is determined as:

ωsim

τ= Ωsim = Ωexp ⇒ τ =

ωsim

Ωexp. (5.2.35)

In order to determine τ , 10 time responses of the force for different base velocities aremeasured and the characteristic frequencies are computed. The results are shown inFig. 5.26 and Fig. 5.27. The mean frequencies result to:

Ωexp = (240 ± 62) Hz and ωsim = 0.1 ± 0.006 , (5.2.36)

what yields the characteristic time as:

τ = (4.7 ± 1.5) · 10−4 s . (5.2.37)

Finally, under the assumption that the contact stiffness is the same for all velocities, thecontact stiffness can be identified as:

kx =m

τ2= m

(

Ωexp

ωsim

)2

= (2.2 ± 1.5) · 106 N/m . (5.2.38)

20 40 600

100

200

300

400

velocity v0 [mm s−1]

freq

uen

cyΩ

exp

[s−

1]

Fig. 5.26: characteristic frequency Ωexp

as a function of the velocity v0

20 40 60

0.06

0.08

0.1

0.12

0.14

velocity v0 [mm s−1]

freq

uen

cyω

sim

Fig. 5.27: characteristic frequency ωsim

as a function of the velocity v0

It shows that the absolute error of measurement is high. This is caused by the highlyvarying frequencies of the experiments as depicted in Fig. 5.26. One reason for this is

112


the changing contact configuration as discussed in section 5.2.1. Firstly, the contactstiffness changes with the load as in the Hertzian contact. Secondly, there is a constantwear that changes the contact area. Thus, the resulting stiffness of (5.2.38) is not morethan an indicator for the magnitude of the real contact stiffness, which obviously changesfor the different test runs. Additionally, it remains unclear if the rest of the parametersis estimated correctly. Furthermore, the parameters κ1 and κ6 used in the simulationsare determined in advance on basis of an educated guess for kx. One can conclude thatthe model at least captures the basic effects of the experiment and shows the correctmagnitudes. Figure 5.28 gives a comparison of the time response of the normalizedmeasured force Fs and the force of the simulation model fs for parameter set 5. It showsthat the amplitudes match relatively good.

2.2 2.22 2.24 2.26 2.28

−4

−2

0

2

4

time t [s]

Fs/√

2µ

mg,

f s/√

2µ

κ4

Fs/√

2µmg

fs/√

2µκ4

Fig. 5.28: time response of the spring force for the experiment and simulation for theparameter set 5. The time of the simulation is computed with the characteristic time τ

Influence of the Spring A more accurate modelling would characterize the connect-ing element between base and specimen as a beam consisting of a stiffness that considersall of the three degrees of motion of the model. It is to be determined whether theseunwanted influences can be neglected. For this purpose the model shown in Fig. 5.29 ischosen that consists of an Euler-Bernoulli beam that is connected to a rigid mass thatis supported by a spring with the contact stiffness kz. As kz ≈ kx ≈ ks applies in thereduction regime, the contact stiffness corresponds to the longitudinal stiffness of thebeam ks as in (5.2.12). The well-known beam theory gives the elastic force FbZ and themoment MbΦ that act from the beam on the mass [106]:

FbZ =3EsIs

l3sw (ls) and MbΦ =

EsIs

lsw′ (ls) . (5.2.39)

Here w (ls) and w′ (ls) give the displacement and the rotation of the beam at x = ls.

113


Φ

x

z, w

Z

b

specimen

kzEs, As, ls

(a) beam model

rs

(b) cross section

Fig. 5.29: (a) beam model of the connection element of the system. (b) cross section ofthe beam

The other important loads are the force FcZ and the moment McΦ of the contact:

FcZ = kzZ = ksZ and McΦ = kzb2Φ = ksb2Φ . (5.2.40)

With (5.2.39), (5.2.40) and (5.2.12) the ratios of the forces and moments yield:

FbZ

FcZ=

3EsIsl3s

w (ls)

EsAsls

Zand

MbΦ

McΦ=

EsIsls

w′ (ls)EsAs

lsb2Φ

. (5.2.41)

As w (ls) = Z and w′ (ls) = Φ as well as As = r2s and Is = π

4 r2s this gives:

FbZ

FcZ=

3

4

(

rs

ls

)2

andMbΦ

McΦ=

1

4

(

b

rs

)2

. (5.2.42)

Finally, with rs = 1 · 10−3 mm, ls = 1 · 10−1 m and b = 1 · 10−2 m this yields:

FbZ

FcZ≈ 10−4 and

MbΦ

McΦ≈ 10−2 . (5.2.43)

Thus, one can conclude that the influence of the connection element on the verticaldisplacement and the rotation can be neglected in the modelling, as the correspondingloads are by orders lower than the contact forces and moments. Therefore, the bendingand torsional-stiffness of the connection element is not considered and it can be modelledas a simple spring.

114

Chapter 6

Conclusion and Outlook

6.1 Conclusion

Two principally different models have been introduced to examine the influence of mi-croscopic slip on the macroscopic behaviour of mechanical systems with friction. Firstly,the oscillatory rolling contact, which corresponds to a Hertzian contact with time vary-ing contact area and pressure distribution. Secondly, the micro-walking machine, whichcorresponds to a rigid slider with several independent contact spots.

6.1.1 Shakedown and Ratcheting in Incomplete Contacts

As generic models for incomplete contacts, the oscillating rolling contacts with constantload regime FT and FN have been introduced. Basic assumptions are dry friction ofthe Coulomb type with constant coefficient of friction µ and linear elastic material be-haviour. Additionally, the systems are assumed to be quasi-static and from an uncoupledtype, i.e. Dundur’s constant β = 0. Thus, variations in the normal force will not induceany tangential displacement and vice versa. The system resembles a tangentially loadedcontact problem from the Hertzian type that is superposed by a slight oscillatory rollingof the upper body. This rolling resembles to a rocking motion, i.e. a back and forthmovement of the point of application of the normal force as in the rocking punch exam-ined by Mugadu et al. [27]. In consequence, the pressure distribution and the contactregion are varied within every cycle of the rolling. In the analysis, the MDR [67], theCONTACT simulation software [88], classical analytical contact mechanics [56, 55] andexperimental test rigs have been used. Depending on the system parameters two distinctcases can occur.

Shakedown In the first few periods the rolling leads to partial slip and increases therigid body displacement of the substrate. In case that the rolling amplitude w and thetangential force fT fall below the so-called shakedown limits, the system reaches a newequilibrium and the centre of the initial contact area remains in a state of stick. Duringthe transient, the traction is increased until a residual force is formed that balances

115

Chapter 6. Conclusion and Outlook

the macroscopic tangential force. This process is referred to as a frictional shakedown[25, 26]. In the experiment, the shakedown state is reached after 10-12 periods of rolling.For all three cases that have been considered, i.e. spherical roller with parallel andperpendicular alignment of load and oscillation and cylindrical roller with parallel align-ment, the shakedown displacement and the shakedown limits have been derived. Theformer describes the new equilibrium state. The latter gives the maximal tangentialforce for a given rolling amplitude for which a safe shakedown occurs and vice versa. Inall three cases it turns out that shakedown is accompanied with a significant reductionof the tangential load capacity. The reduction in all three cases is approximately:

fT,lim ≈ 1 − wlim . (6.1.1)

This effect must be considered in the design of force locked connections in which thecontact is incomplete and the pressure distribution varies in the described manner. Com-parison of the three cases shows that spherical contacts can bear higher tangential loadsthan cylindrical ones. In addition, the parameter range of the shakedown displacementis narrower for the cylindrical case, i.e. the final shakedown displacements differ onlyslightly for different fT and w.

Ratcheting In case that the shakedown limits are exceeded, an effect called ratchet-ing occurs. The alternating slip processes cover the complete initial contact area andshakedown is no more possible. Depending on the actual rolling direction one side ofthe contact sticks while the other slips. This accumulated displacement results in a rigidbody motion referred to as ratcheting or walking [27, 28, 29]. Approximation functionsfor the incremental slip per period as a function of the tangential load and the rollingamplitude have been derived. These are in good agreement with the experimental resultsand show qualitatively good agreement with the results for the walking of the rockingpunch [27]. The ratcheting effect can be used for the generation and control of smalldisplacements in case that an increase of the tangential loading is not possible or highaccuracy is needed as in MEMS. Comparison shows that the incremental displacementis highest for the spherical contact with perpendicular setting of load and rolling andlowest for the cylindrical setting.

The results of the different methods show good agreement. The MDR [67] enables todeduce the analytical form of the shakedown state and the shakedown limits. Thiswould have been almost impossible using classical contact mechanics as for instance theCatteneo-Mindlin approach [59, 60] or the Ciavarella-Jager principle [58, 62]. This isbecause both, contact area and traction distribution are not known and are the productof a transient process. However, comparison with the experiments and the 3-D CON-TACT simulation indicates that the MDR results are only qualitatively correct. Theslight deviation that occurs is caused by the missing rotational symmetry of the contactconfiguration after shakedown, what violates one basic assumption of the MDR. Never-theless, the results show that the MDR has again proven to be a very suitable methodfor the analytical description and simulation of frictional contacts.

116


6.1.2 Dynamic Influences on Sliding Friction

The so called micro-walking machine is introduced in order to study the system behaviourin case that the tangential load is sufficient to induce gross sliding. It consists of a rigidbody with several contact spots that are assumed to be linear elastic and independentfrom one another. Again, dry friction of the Coulomb type is assumed with a constantcoefficient of friction µ and no distinction between static and kinetic case. The systemtakes into account spatial variation of stick- and slip zones and contact forces, therebyenabling micro-walking of the specimen. This characterizes a dynamic mode in whichthe system travels most of the forward motion while the normal forces in the contactsare low in comparison to the average value. This causes a reduction of the effectivecoefficient of friction, i.e. a lower apparent macroscopic frictional resistance. For theanalysis numerical simulations and an experimental rig have been used.

Parameters of Influence In a first step, parameter combinations are identified forwhich a significant reduction occurs. The results indicate that a reduction is inducedif the stiffness of the excitation and the contact stiffness are comparable in size. Inaddition, the friction decreases with increasing velocity, increasing microscopic frictionand increasing coupling between rotational moment and friction forces. These resultsare consistent with those gained with more or less comparable models [10, 47, 50].

Vanishing Frictional Resistance The maximal reduction is 98 % in theory and73 % in the experiments. A further analysis of the results shows that the reductionis caused by self-excited oscillations that are induced by the coupling of the differentdegrees of motion. The excitation causes a rotation of the rigid body which increasesthe contact forces and induces oscillations in the vertical direction. This instability ischaracterized by a microscopic jumping of the rigid body with released contacts that isin strong correlation with the lateral motion: low or zero contact forces coincide witha fast forward motion of the rigid body. In addition, the alternation between storageand motion phase is identified as the prerequisite for the characteristic jumping and fastforward movement. The reduction increases with increasing velocity, increasing leverarm of the excitation and with increasing microscopic friction as all of these effects sup-port the instability, i.e. increase the amplitude of the vertical oscillations. However,the vertical amplitudes in the reduction range are of the order of the static deflection,i.e. the self-excited oscillations are from a microscopic character. Taken together, themodel shows that micro-vibrations play an important role for the dynamic influenceson the effective frictional resistance of systems that exhibit apparently smooth sliding.In these systems, the experimentally observed dependency of the frictional resistanceon dynamic quantities might be explained by microscopic effects that are influenced bymacroscopic system features such as the stiffness or the geometry. One example for thisbeing the rate weakening of the friction coefficient. The model shows that this effectcan be explained to some extent by dynamic instabilities that simply increase with thesliding velocity. This is particularly important for the design of tribometers.

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The results of the simulations give a clear but also strict guideline for the parameterrange that needs to be adjusted if one wants to induce this reduction through dynamicinstabilities. Both, the design-phase of the experimental rig and the experiments itselfshow that this parameter range is narrow and hard to reach. For instance, matchingthe non-dimensional parameters in reality is very challenging. Furthermore, a changeof the scaling of the model would change the non-dimensional parameters, due to crossinfluences and some constant parameters as for instance the gravitational acceleration.Thus, it remains vague to what extent the proposed effect occurs in practical systems.However, the results provide a solid starting point for further research and developmentof applications in technical systems.

6.1.3 The Interplay of System Dynamics and Friction

The first model explains the failure of force locked connections under the influence ofoscillations in the pressure distribution. The second model shows that micro-vibrationscan be responsible for the dynamic influences on the effective frictional resistance ofsystems that exhibit apparently smooth sliding. Although both systems are relativelydifferent in terms of basic assumptions (nominally static vs. dynamic) and type of anal-ysis (analytic derivations vs. numerical experiments) the results are fairly connected toeach other. As shown in Fig. 6.1, the different phenomena can be imagined as neigh-bouring scenarios. The range of scenarios starts on the left with the shakedown effect.In this range, a system that is subjected to oscillating loads will reach a new equilibriumafter some periods, i.e. it will remain stable as long as both tangential force and oscil-lation amplitude fall below the shakedown limits. If these are exceeded, the system willenter the next scenario. Ratcheting starts which is an accumulation of micro-slip thatalternately affects the whole contact area and leads to a continuing rigid body motion. Ifthe instability limit is exceeded, the system enters the next scenario: the micro-walkingrange. Here, self-excited oscillations occur that lead to temporarily released contacts.The instability limit depends on different system properties such as the sliding velocity,stiffness, mass and geometry.

v0w

shakedown ratcheting micro-walking

Fig. 6.1: scenarios in mechanical systems with friction and oscillations

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The consequences of the different effects are very similar. In the case of shakedownoccurs a significant reduction of the maximal tangential load capacity. In consequence,the force that is needed to induce a rigid body motion in the case of ratcheting is lowerthan expected, i.e. the effective frictional resistance is reduced. This is the same effect asin the case of micro-walking. Thus, as stated in the introduction of this work all of theseeffects strongly influence the macroscopic behaviour of the system although they occurat the micro-scale. In the model for shakedown and ratcheting in incomplete contacts,the oscillations are externally driven. In contrast, in the micro-walking machine theoscillations are self-excited. Consequently, throughout this work the different effects areconsidered separately. However, a mechanical system with friction that is subjected tooscillations in the contact area or contact stresses can successively or simultaneouslyexhibit all these effects. Therefore, future research should combine the two models togive further insights to the subject of dynamic influences in mechanical systems withfriction.

6.2 Outlook

In addition to a possible combination of the two models proposed in this work, there existseveral other promising extensions that are briefly described in the following section.

6.2.1 Shakedown and Ratcheting in Incomplete Contacts

Shakedown in Contacts of Rough Surfaces Since all technical surfaces are roughto some extent, one interesting extension of the work is to consider shakedown and ratch-eting in the contact of rough surfaces. Starting point is the tangentially loaded contactbetween an elastic half-space and a rigid rough indenter. Instead of oscillatory rolling oncan consider rocking of the contacting bodies that is induced by a moving normal forceas in the rocking and walking punch of Mugadu et al. [27]. The MDR enables to mapthe initial 3D system to a one-dimensional system of independent elements [67] and hasproven to be a suitable instrument for the modelling of fractal rough surfaces [70, 71].For this reason, one can expect that also shakedown and ratcheting in rough surfacescan be simulated using the MDR.

Shakedown for Further Contact Geometries Further research is needed to ex-amine other forms of smooth rolling bodies as cones or ellipsoids. Although it remainsvague whether an analytical description of the shakedown limits of these systems can beachieved, it is certainly possible to conduct 3D simulations using the CONTACT modeland to conduct experiments using the rolling body test rig.

Interaction of Parameters Especially for technical applications it would be inter-esting to consider the interaction of different parameters. For example, the oscillatoryrolling might be superposed by varying normal and tangential forces. Or the influenceof the alignment of load and oscillation for angles between 0 and π/2 can be considered.

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6.2.2 Dynamic Influences on Sliding Friction

Extended Experiments As the theoretical maximal reduction of the frictional re-sistance is not reached, further development of the experimental test rig is needed inorder to match the maximal range of the non-dimensional parameters. One possibilityis to identify other maximal parameter combinations that are easier to reach in practice.Another possibility is to attach some kind of guide-way such that the two-dimensionalcharacter of the theoretical model can be reproduced. For this, a plane substrate insteadof a prism shaped substrate needs to be used.

Contribution of Instability Mechanisms The simulations show that several mech-anisms contribute to the reduction effect: vibrations in the normal direction, synchro-nization of tangential and normal motion and characteristic alternation between storageand motion phase. Attempts should be made to reveal which of the proposed effectscontributes mostly to the reduction. For this purpose, models and experiments must bedesigned where certain effects can be switched off. For instance, one can avoid jump-ing of the specimen using a vertical guidance in order to examine the influence of therotation.

Practical Guidelines The origin for the work on the dynamic influences of the fric-tional resistance is the poor reproducibility of measurements of friction coefficients.Therefore, the results should be revised in order to give practical guidelines for thedesign of tribometers. This can be followed by analysis and improvement of an existingtribometer with regard to its dynamic properties.

Measurement of Normal Oscillations The only dynamic quantity that is measuredin the experiments is the time response of the spring force at the base. Although, thetime responses of the spring forces of simulation and experiment match relatively good,it would be an important extension to measure the normal translation of the rigid bodyin order to compare it to the simulation results. Especially, since in accordance to manyother authors [39, 40] the normal oscillations are responsible for the reduction effect.The experimental set-up makes it easy to measure the normal displacement as the meanposition of the rigid body is constant and the substrate is moved. For this, the laservibrometer used in the rolling body experiments can be applied to the micro-walkingmachine test rig.

Parameters The DoE analysis indicates that certain parameters deserve further anal-ysis as they have a significant impact on the frictional resistance. For instance, increasingratio of height and width of the specimen increases the friction. For this purpose, alsoinstability analysis can be used, where one considers perturbations of the steady slidingsolutions.

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[117] J. J. Kalker. A Fast Algorithm for the Simplified Theory of Rolling Contact.Vehicle System Dynamics, 11(1):1–13, 1982.

[118] J. J. Kalker. Three-Dimensional Elastic Bodies in Rolling Contact. Springer,October 1990.

[119] M.P. Allen and D.J. Tildesley. Computer Simulation of Liquids. Clarendon Press,Oxford, 1987.

129

Appendix A

Statistics

A.1 Analysis of the Experiments

A.1.1 Specification of Measurement Results

Experimental measurements are always subjected to some statistical deviations, whichare caused, among other things, mainly by imperfections of the instruments and theiroperation. The specific value xi of a measurement i differs from one another and thusscatters around the mean value x, which is defined as the arithmetic mean of the par-ticular measurements:

x =1

n

n∑

i=1

xi . (A.1.1)

With increasing number of measurements, the arithmetic mean approaches the so-calledexpected value xe. In order to evaluate the uncertainty of measurement, the standarddeviation of the mean value is considered, which indicates how far the mean value ofmeasurement differs from the true value xe :

σx =1√n

√

√

√

√

1

n − 1

n∑

i=1

(xi − x)2 . (A.1.2)

For n → ∞ one can determine with the probability density h(x) that the true value xe

lies within the range x ± σx with a probability of 68.3%. This range is defined as theconfidence interval on the confidence level 68.3%. In case of only a few measurementvalues n, one has to use the so-called student-t distribution for the probability densityh(x). This is considered by the correction term t that is tabulated for instance in theDIN norm 1319-3 for different numbers of measurements n and for various confidenceintervals [107]. A measurement result is then specified as follows:

x = x ± tσx , (A.1.3)

where t is taken from tables. Unless otherwise noted, the measurement results in thepresent work are given with confidence intervals on the confidence level 68.3%. In case

130

Appendix A. Statistics

that a certain measurement variable y is a function G1 of an arbitrary number m ofuncorrelated variables xk the uncertainty of measurement follows from the so calledGaussian error propagation law that is also defined in the DIN norm 1319-4 [108]:

σy =

√

√

√

√

m∑

k=1

(

∂G

∂xk

)2

σ2xk

. (A.1.4)

A.1.2 Identification of Parameters

Not all parameters of the experimental rigs are known or can be measured directly.Examples include the coefficient of friction µ, the elastic modulus of the substrate of therolling body test rigs E2 and the mapping parameter κ that occurs due to the violatedrotational symmetry. In order to determine the unknown values the so called methodof least squares is used. In the first step, the sum of the squared deviation between themeasured variable yi and the model function f (xi, αj) is computed:

Sj =n∑

i=1

(f (xi, αj) − yi)2 . (A.1.5)

Here αj denotes a specific set j of a number of model parameters in the model func-tion and n is the number of observations or measurements. In the second step, theparameter set j for which the corresponding Sj is minimal is chosen for determinationof the unknown parameters. The two basic assumptions for this approach are normallydistributed measurement deviations of yi and a correct measurement of the independentvariable xi. In practice this means that the measurement deviations of xi must be smallcompared to those of yi [109]. Linear model functions f(xi, αj) enable the analytical de-termination of both the minimum and the parameters αj . Otherwise iterative methodsare used [110]. However, in the present work mostly non-linear functions derived fromphysical principles are used for f (xi, αj), as for instance the static displacement of thesubstrate Ustat (FT , α) with α = (E, µ). Often, only one or two parameters occur in thefunctions. Thus it is useful to compute Sj for a sufficiently high resolution of αj . Theminimum can then easily be detected.

A.1.3 Curve Fitting and Regression

Curve fitting and regression are widely used instruments in many areas, particularlyin econometrics, where a good introduction is given by Woolridge [111]. The aim isto construct a smooth function that approximately fits the given data xi and yi. Forinstance in a linear regression analysis the starting point is the model function:

y = αx + β . (A.1.6)

The coefficients α and β have to be determined such that the model function reflects therelation between x and y as good as possible. For this the least square method is used.

1Here G can be a known function or a guessed model function.

131


Based on the estimated parameters α and β one can calculate the fitted value yi. Thedeviation between the fit yi and each measured value yi is called the residual δyi:

yi = yi + δyi . (A.1.7)

The goodness of the fit is determined by several measures. Besides others, the explainedsum of squares (SSE) and the residual sum of squares (SSR) are important:

SSE =n∑

i=1

(yi − y)2 , (A.1.8)

SSR =n∑

i=1

δy2i , (A.1.9)

where y is the mean of all measured values.

Tab. A.1.1: quality of the curve fitting results

reference fitted model function SSE R2

(3.2.17) fT = 1 − wusd − (1 − usd)3/2 3 · 10−7 1

(3.3.2) ηfit = 4.77√

ustat 0.257 0.986

(3.3.4) ∆u =

√

2(

1 − (1 − fT )2/3

)

(w − wlim) 0.011 0.989

(3.4.8) fT = 1 − 1.1wusd − (1 − usd)3/2 5 · 10−4 0.999

(3.4.15) ∆u =

√

5(

1 − (1 − fT )2/3

)

(w − wlim) 0.048 0.975

(3.5.11) Ustat = U0

(

1 −(

1 − FTµFN

))0.923 · 10−5 1

(3.5.12) U0 = µ 2FNπE∗L

(

12 (1 + ln (4)) + ln

(

1.8E∗

G∗

La

))

4 · 10−4 1

(3.5.26) usd = 1 − (1 − fT )0.92 + 0.3fT 7 · 10−2 0.999

(3.5.28) wlim = 1 − f0.73T,lim 2 · 10−2 0.999

(3.5.29) ulim = f0.87T,lim 5 · 10−5 0.999

(3.5.30) ∆u = 0.64fT (w − wlim) 2 · 10−3 0.95

SST and SSR define the coefficient of determination R2 which is the ratio of the ex-plained variation and the total variation:

R2 =SSE

SSE + SSR. (A.1.10)

The ratio R2 is interpreted as the fraction of the sample variation in y that is explainedby x. In case of a perfect fit all data points would lie on the model function, which yieldsR2 = 1. A value of R2 equal to zero indicates a poor fit. Table A.1.1 gives an overviewof the different fits together with the SSE and R2 values.

132


A.1.4 Correlations

The correlation coefficient measures the strength of the linear correlation of two variablesx and y. One of the most common coefficients is the Bravais-Pearson product-momentcorrelation r, which is a dimensionless index that is invariant to linear transformationof either variable [112]:

r =

∑ni=1 (xi − x) (yi − y)

√

∑ni=1 (xi − x)2∑n

i=1 (yi − y)2. (A.1.11)

Here x and y denote the mean value of a sample with n observations. Consider a diagramin which xi depicts the horizontal coordinate and yi depicts the vertical coordinate. Themore the distribution of values of xi and yi resembles a straight line, the closer is r to 1or -1 in case of a negative slope. It is important to know that the case r ≈ 0 does notindicate no correlation but only indicates no linear correlation. In a coarse raster onecan classify the correlation as [113]:

|r| < 0.5 weak correlation ,0.5 ≤ |r| < 0.8 medium correlation ,0.8 ≤ |r| strong correlation .

(A.1.12)

But this raster strongly depends on the variable examined. Therefore, correlation coef-ficients are particularly useful as comparative measurements. They are rather from anexploratory and descriptive character. The following form of (A.1.11) is very useful fora revolving calculation of r as needed for simulations on the GPU:

r =

∑ni=1 xiyi − nxy

√

(∑n

i=1 x2i − nx2

) (∑n

i=1 y2i − ny2

)

. (A.1.13)

A.2 Design of Experiments

The term design of experiments (DoE) describes a specific method used for an efficientplanning and execution of experiments with several parameters of influence [114, 115].The intended area of application are experiments in the real world, where the experimen-talist often lacks deep knowledge about the real underlying mechanisms of the examinedsystem. The DoE then helps to fill this black box and enables to develop model func-tions. In the case of the micro-walking machine exists a precisely known numericalmodel. Thus, it is not necessary to consider the experimental principles of replication,randomization and blocking because the experimental conditions do not change. How-ever, one can make good use of the evaluation methods. In the first step, the overallnumber of parameters is divided into two groups: factors whose influence is studied andall other parameters that are held constant. The different adjustments of the factorsare called levels. In case of the basic principle of the DoE, the so-called full factorial

133


experimental schedule, every factor is varied and all possible combinations are taken intoaccount. For n levels and k factors, the full-factorial schedule gives:

m = nk (A.2.1)

different combinations of parameters. For example, if one considers two factors a, b andtwo levels for each factor, this yields 22 = 4 points in the parameter space as shown inFig. A.1 (a). The experiment gives four different results y. The so-called main-effecty is a measure for the influence of a specific factor and is given by the mean for allcombinations in which this factor is held constant. For example the main effect for thefactor a = a1 is given as:

ya1 =1

2(y (a1, b1) + y (a1, b2, )) . (A.2.2)

The so called main-effect plot summarizes all main effects as in Fig. A.1 (b). In addition,the interaction-effect plot shows the interaction between the different factors, as shownin Fig. A.1 (c). It gives the influence of a specific factor, b in this case, while the otherfactor is kept constant.

ab

(a1, b1) (a2, b1)

(a2, b2)(a1, b2)

(a) parameter space

a1 a2

y1

y2

y

(b) main effect

b1 b2

a = a1

a = a2

y

(c) interaction effect

Fig. A.1: design of experiments (DoE). (a) parameter space. (b) main effect plot. (c)interaction effect plot

The drawback of the full factorial experimental schedule is the large number of factorlevel combinations in comparison to other methods of the DoE. However, as every ex-periment is only conducted once, the full-factorial schedule is used in the analysis of theparameters of influence of the micro-walking machine.

134

Appendix B

Numerical Methods

In order to carry out simulations of the different systems, several numerical modelsare used that are briefly discussed in the following section. Important algorithms areillustrated in pseudo-code-notation. Please refer to the given references for more detailedinformation on the methods used in this work.

B.1 Incremental MDR Simulation

One of the most important properties of the MDR model is the decoupling of the degreesof freedom of the elements of the foundation. Consequently, the numerical effort of theMDR system is of the order O = n, where n gives the numbers of elements, i.e. degreesof freedom. In comparison, the numerical effort of the boundary element method (BEM),which is widely used in contact mechanics, is of the order O = n4. In this way, the MDRallows a very fine discretization with low computational complexity and enables detailedparameter identifications [67].The oscillating rolling contact as introduced in section 3.1 is simulated with the MDRusing a quasi-static incremental approach. Here, the rolling path W is divided intoincremental steps with step-size ∆W . The displacement of the substrate U is thencomputed in every time step on basis of the equilibrium of forces. The simulation isdivided into two parts, namely the rolling part and the deflection part.

Rolling Part On basis of the half-space assumption rolling only changes the normaldisplacement Uz. Thus, for an arbitrary spring at position s for step k this yields:

Uz (s, k) = d − (s ∓ k∆W )2

R, (B.1.1)

fz (s, k) = ∆kzUz (s) . (B.1.2)

Here −k∆W in (B.1.1) simulates forward rolling and +k∆W backward rolling. Thecorresponding contact region for step k is shifted as:

−a ± k∆W ≤ s ≤ a ± k∆W . (B.1.3)

135

Appendix B. Numerical Methods

Displacement Part Firstly, the displacement is slightly increased so that the testdisplacement Ux and the test-tangential force fx read:

Ux (s, k) = Ux (s, k − 1) + ∆U , (B.1.4)

fx (s, k) = ∆kxUx (s, k) , (B.1.5)

U (k) = U (k − 1) + ∆U . (B.1.6)

Here Ux (s, k − 1) and U (k − 1) denote the old spring deflections respectively the oldrigid body displacement. Given the test force, the stick and slip regions are identifiedon basis of the traction bound. The new displacements are then computed as:

fx (s, k) ≤ µfz (s, k) ⇒ stick-region: ⇒ Ux (s, k) = Ux (s, k − 1) + ∆U ,

fx (s, k) > µfz (s, k) ⇒ slip-region: ⇒ Ux (s, k) = µE∗

G∗Uz (s, k) .

(B.1.7)

Consequently, the new tangential forces result to:

fx (s, k) = ∆kxUx (s, k) . (B.1.8)

In principle, rolling initially decreases the tangential force in the contact Fx (k) =∑

fx (s, k). This leads to an imbalance of the forces what in turn increases the dis-placement. Thus, the displacement step is repeated until:

Fx (k) ≥ FT (1 − ε) , (B.1.9)

where ε denotes the relative accuracy of the calculation. This loop is repeated forevery single rolling step as shown in Fig. B.1, where the stepwise incremental simulationscheme is illustrated. In principle, the proposed scheme can be adapted to a system thattakes into account the inertia of the contacting bodies, i.e. the macroscopic dynamics,as described in chapter 13 in the book of Popov and Heß[67].

k periods

W

1 2

step k

repeat

rolling part displacement part

Fx ≥ FT (1 − ε)

Uz = d − (s∓k∆W )2

R Ux = Ux + ∆U

Fig. B.1: stepwise incremental simulation scheme for the oscillating rolling contact

136


B.2 CONTACT Simulation

In order to conduct a three-dimensional, quasi-static simulation of the oscillating rollingcontacts, the well-known CONTACT software package is used, whose first version wasreleased 1982. The program has since been continuously developed by the companyVorTech BV and its fields of application have been expanded. It is freely availablefor academic use. The program implements the theories of rolling contacts by Kalker[116, 117, 118] and is based on the boundary element method (BEM). Here, the elasticityequations for the interior of the contacting bodies are transferred to equations for theirbounding surfaces. Under the half-space assumption, the main resulting equation is thedisplacement and stress relation as given by the potential functions of Boussinesq andCerruti, see section 2.1.1 and section 2.1.2. The code uses constant element discretizationand the potential contact area is split into mx×my rectangular elements of size ∆x×∆y.The contact problem is solved using nested iteration processes. In case of a decoupledsystem the normal problem is solved first. As this gives the contact area and the tractionbound, the tangential problem is solved subsequently. Here, the solver initially guessesa potential subdivision of stick- and slip area which is then updated on basis of contactconditions [88]. The iteration processes are based on a Newton-Raphson procedure, that(in this case) determines the indentation d and the tangential shift U on basis of theprescribed forces FN and FT . Once the requested relative accuracy ε is reached, theiteration process does not improve the solution any more. For tangential forces closeto the maximum CONTACT lacks robustness [88] due to the non-linear dependenceof total forces on creepages [55]. A so called world-fixed coordinate system is used inwhich the rolling body moves with velocity v in the x-direction and the particles in thecontact area more or less stay at the same coordinate. The geometry of the contactingbodies is entered using a so-called Hertzian approach. Here, the initial distance betweenthe undeformed surfaces h (x, y) is entered directly. The rolling is then simulated as anincremental stepwise shift of the profile with ∆W being the step-size. In order to avoidinaccuracies or convergence problems ∆W should be comparable to the grid-size in therolling direction ∆x [88]. Because CONTACT has only a rudimentary graphical userinterface (GUI), a MATLAB script is used to produce the text-based input files.

y x

z h0 (x, y)

∆x∆y

(a) spherical contact

y

x

z h0 (x, y)

∆x

∆y

(b) cylindrical contact

Fig. B.2: numerical model of the CONTACT software. (a) spherical: discretization alongthe x- and y-direction. (b) cylindrical: just one line of elements along the x-direction

137


Spherical Rolling Contact For a rolling sphere the initial profile h0 (x, y) reads:

h0 (x, y) =x2

2R+

y2

2R. (B.2.1)

In case of w ‖ fT , the actual profile after k rolling steps reads:

hk (x, y, k) =x2

2R− k∆W

Rx +

(k∆W )2

2R+

y2

2R. (B.2.2)

For a mutual verification, the exact parameters of the experimental test rigs described insection 5.1.1 in Tab. 5.1.1 and in section 5.1.2 in Tab. 5.1.2 have been used. The numberof elements varies with the amplitude due to the world fixed approach. For instance, witha grid-size of ∆x, ∆y = 0.2 mm and an incremental step size of ∆W = 0.2 mm result3050-4200 discretization elements. For the shakedown effect 10 periods of rolling aresimulated resulting in 241-681 computation steps or 2 − 4 h running time on a standardPC for each single parameter combination of fT and w .

Cylindrical Rolling Contact For a rolling cylinder the initial profile h0 (x, y) reads:

h0 (x, y) =x2

2R, (B.2.3)

and the actual profile after k rolling steps reads:

hk (x, y, k) =(x + k∆W )2

2R. (B.2.4)

The cylindrical shaped roller with length L is modelled as a truncated 3D problem.This means that there is just one row of elements and all the contact quantities suchas pressure or traction are constant along the y-direction. The numerical problem isthus two-dimensional1 as depicted in Fig. B.2 (b). Again, the exact parameters of theexperimental test rig described in section 5.1.3 in Tab. 5.1.3 have been used. The numberof elements varies with the amplitude due to the world fixed approach. For a grid-sizeof ∆x = 0.02 mm, ∆y = L and an incremental step size of ∆W = 0.02 mm result 220-307 discretization elements. For each combination of fT and w 10 periods of rollingare computed, resulting in 881-2641 computation steps or 1 − 2 min running time on astandard PC per combination.

1The problem still resembles to the loading of a three-dimensional elastic half-space.

138


B.3 Time Integration

Newton’s seconds law yields the general initial value problem for the motion of a me-chanical system as:

x (t) =F (t)

mand x (0) = x0, x (0) = v0 . (B.3.1)

The motion of the system can be computed using a stepwise numerical integrationscheme. For the simple Euler method the solution is based on the difference quotients:

x =x (t + ∆t) − x (t)

∆tand x =

x (t + ∆t) − x (t)

∆t(B.3.2)

what gives [97]:

(

x (t + ∆t)x (t + ∆t)

)

=

(

x (t)x (t)

)

+ ∆t

(

x (t)F (t)m

)

, (B.3.3)

where ∆t denotes the time step. In rigid body dynamics one often uses multistep meth-ods as the famous Runge-Kutta method [97]. Most of these classical procedures requireknowledge of the exact time course of the force F (t). This might be one reason for thewidespread use of velocity dependent friction laws. However, in the present model theforces in the contact are a direct result of the spring displacements, thus the schemeproposed in (B.3.3) cannot be used. But this situation almost exactly corresponds tothe situation in molecular dynamics (MD) where the forces that act on the moleculesare a direct result of their actual relative position. This leads us to the commonly usedVelocity-Verlet algorithm whose scheme is as follows [119]:

x (t + ∆t) = x (t) + ∆tx (t) +1

2(∆t)2 F (t)

m, (B.3.4)

x (t + ∆t) = x (t) +1

2∆t

(

F (t)

m+

F (t + ∆t)

m

)

. (B.3.5)

The scheme enables to determine the new displacements first which then serve as inputfactors for the new forces F (t + ∆t).

B.4 Contact Models for the Micro-Walking Machine

Using the separation of space scales principle [65] as discussed in section 2.3.1, thecontact spots of the micro-walking machine are modelled as linear springs as describedin the sections 2.3.3 and 4.1.1. As the spring deflections in the normal and tangentialdirections directly depend on the actual motion of the rigid body, i.e. x, z and ϕ, a casedistinction is being used to determine them. Here the subscript (.)1/2 denotes the twocontact spots, thus every case distinction is carried out separately for the two spots onthe left (1) and the right (2) of the specimen.

139


Two-dimensional Contacts In the model described in section 4.1.1 each spring hasone normal deflection uz1/2 and one tangential deflection ux1/2 aligned parallel in the z-and x-direction as in Fig. B.3 (a). Firstly, a test normal deflection uz1/2 is computed:

uz1/2 (t) = z (t) ± κ−13 ϕ (t) , (B.4.1)

and it is checked whether a specific spring is released from the substrate:

uz1/2 (t) > 0 ⇒ spring in contact ⇒ uz1/2 (t) = z (t) ± κ−13 ϕ (t) ,

uz1/2 (t) ≤ 0 ⇒ spring is released ⇒ uz1/2 (t) = 0 .(B.4.2)

After that, the tangential deflections ux1/2 are computed via a second case distinction.For a sticking contact, the deflections are directly determined by the change of rigid bodymotion between the two time steps, i.e. ∆x, ∆z and ∆ϕ. Thus, the test deflections arecomputed firstly:

ux1/2 (t) = ux1/2 (t − ∆t) + ∆x + ∆ϕ , (B.4.3)

where ux1/2 (t − ∆t) denotes the old deflections. If this test deflection exceeds the frictionbound µκ2uz1/2 (t), the contact slips. Consequently, the case distinction for the newtangential deflections reads:

|ux1/2 (t) | ≥ µκ2uz1/2 (t) ⇒ spring slips ⇒ ux1/2 (t) = µκ2uz1/2 (t) sign(

ux1/2 (t))

,

|ux1/2 (t) | < µκ2uz1/2 (t) ⇒ spring sticks ⇒ ux1/2 (t) = ux1/2 (t) .

(B.4.4)

xz

specimen

substrate

(a) 2D contact

y

zs

n

αspecimen

substrate

(b) 3D contact

xs

n

α

contact

µκ2un

ux

us

√

u2x + u2

s

(c) topview

Fig. B.3: (a) contact model of initial model with linear spring. (b) contact model of theextended model with additional spring deflection us. (c) top view of the surface of thesubstrate of the extended contact model with friction bound µκ2un

Three-dimensional Contacts The contact geometry of the experimental specimenas introduced in section 5.2 differs slightly from the numerical model described in sec-tion 4.1.1. In order to identify appropriate parameter ranges and to enhance compa-rability an extended contact model is being used. The prism shaped substrate addsan additional deflection us1/2 to each spring. As shown in Fig. B.3 (b) and (c) us1/2 isaligned perpendicular to the normal deflection un1/2 as well as to the tangential deflectionux1/2. The tangential stiffness of the two directions is the same as in the two-dimensional

140


contact, i.e. kx in the dimensionful model. The normal direction of the contact is nowinclined by 45 to the z-direction and the test normal deflection yields:

un1/2 (t) =

√2

2

(

z (t) ± κ−13 ϕ (t)

)

. (B.4.5)

The first case distinction reads:

un1/2 (t) > 0 ⇒ spring in contact ⇒ un1/2 (t) =√

22

(

z (t) ± κ−13 ϕ (t)

)

,

un1/2 (t) ≤ 0 ⇒ spring is released ⇒ un1/2 (t) = 0 .(B.4.6)

Again, in case of sticking contacts, the deflections directly follow from the change ofmotion between the time steps, i.e. ∆x, ∆z and ∆ϕ. This yields the tangential testdeflections as:

ux1/2 (t) = ux1/2 (t − ∆t) + ∆x + ∆ϕ , (B.4.7)

us1/2 (t) = us1/2 (t − ∆t) +

√2

2

(

∆z − κ−13 ∆ϕ

)

. (B.4.8)

In case of slipping contacts, the friction bound µκ2un1/2 only gives the norm of thetangential deflections. Thus, one has to determine the angle α1/2 between ux1/2 andus1/2 as shown in Fig. B.3 (c). With the test deflections ux1/2 and us1/2, the casedistinction for the three-dimensional contacts at time t thus finally reads:

√

u2x1/2 + u2

x1/2 ≥ µκ2un1/2 ⇒ spring slips ⇒ α1/2 = arctan(

| us1/2

ux1/2

|)

,

⇒ ux1/2 = µκ2un1/2 cos(

α1/2

)

,

⇒ ux1/2 = µκ2un1/2 sin(

α1/2

)

,√

u2x1/2 + u2

x1/2 < µκ2un1/2 ⇒ spring sticks ⇒ ux1/2 = ux1/2 ,

⇒ us1/2 = us1/2 .

(B.4.9)

141

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